# Is Twitter Enough? Investigating Situational Awareness in Social and Print
Media during the Second COVID-19 Wave in India

Ishita Vohra, Meher Shashwat Nigam, Aryan Sakaria, Amey Kudari, Dr. Nimmi
Rangaswamy  International Institute of Information and Technology, Hyderabad,
India
{ishita.vohra, meher.shashwat, aryan.sakaria<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

The COVID-19 pandemic required efficient allocation of public resources and
transforming existing ways of societal functions. To manage any crisis,
governments and public health researchers exploit the information available to
them in order to make informed decisions, also defined as situational
awareness. Gathering situational awareness using social media, has been
functional to manage epidemics. Previous research focused on using discussions
during periods of epidemic crises on social media platforms like Twitter,
Reddit, or Facebook and developing NLP techniques to filter out
important/relevant discussions from a huge corpus of messages and posts.
Social media usage varies with internet penetration and other socio-economic
factors, which might induce disparity in analyzing discussions across
different geographies. However, print media is a ubiquitous information
source, irrespective of geography. Further, topics discussed in news articles
are already ‘newsworthy’, while on social media ’newsworthiness’ is a product
of techno-social processes. Developing this fundamental difference, we study
Twitter data during the second wave in India focused on six high-population
cities with varied macro-economic factors. Through a mixture of qualitative
and quantitative methods, we further analyze two Indian newspapers during the
same period and compare topics from both Twitter and the newspapers to
evaluate situational awareness around the second phase of COVID on each of
these platforms. We conclude that factors like internet penetration and GDP in
a specific city influence the discourse surrounding situational updates on
social media. Thus, augmenting information from newspapers to information
extracted from social media would provide a more comprehensive perspective in
resource-deficit cities.

###### Index Terms:

Situational Awareness, Twitter, Print Media, Data Mining, Web Mining

††publicationid: pubid: IEEE/ACM ASONAM 2022, November 10-13, 2022
978-1-6654-5661-6/22/$31.00 © 2022 IEEE

##  I Introduction

We explore the application of situational awareness [4] as a concept in
medical emergencies like the COVID pandemic, combining knowledge of the
dynamic present with pre-existing wisdom to determine likely events that can
occur in the near future. Having a perception of the immediate environment and
the information around it leads to levels of situational awareness that help
people and organizations to respond effectively. Studies have harnessed social
media to generate situational awareness in times of need.
During the second wave of the COVID-19 pandemic in India, people took to
Twitter for assistance, especially crowd sourcing medical resources such as
oxygen cylinders, hospital beds, and other medical requirements. Despite
Twitter not being the go-to social media platform for Indians, the number of
tweets from India rose to about 7 times during the peak of the pandemic, in
April-May of 2021 as compared to February-March[12]. Users took to Twitter to
broadcast medical crises in their localities, information relating to medical
assistance, and logistical issues that the second wave was causing.
Information of this kind has been analyzed in the past to create situational
awareness for specific events and catastrophes. These studies allegedly assume
Twitter to be a comprehensive source of harnessing public discourse around
important topics discussed on any subject in real-time.[17, 18, 19]. This
reasoning, in our opinion, maybe acceptable for geographies where the majority
of people use social media as a primary of communication. However, cities with
limited access to the internet have irregular access to real time information.
Consequently, in internet constrained areas, sources such as print media could
likely offer a more comprehensive public discussion. Prior work has suggested
print media as one of the key avenues for taking information to the masses
about the spread of the pandemic, institutional curtailment efforts among
other critical issues.[25].
Our study focuses on the following:

  * •

It investigates data-augmentation over and above the twitter platform from
print media sources in creating situational awareness of an event

  * •

We specifically look into social media and print media and their combined
augmentation during the second wave of the COVID crisis. To our knowledge,
there is no prior research combining the two media formats to this end.

  * •

We explore Twitter data and the Times of India (the largest English print
newspaper in India) and Hindustan Dainik (a newspaper in the Hindi Language)
as our news sources, to understand public discourse surrounding situational
updates.

  * •

Using the data collected, we compare data augmentation in multiple cities with
differing socio-economic factors.

To summarize our main contribution is as follows: We use data from Twitter and
traditional print news media, to understand the diversity of information that
each of them offered during the second wave of COVID pandemic using NLP and
manual techniques. Further, we augment the data across the two media for six
major Indian cities. These cities are diverse socio-demographically and in
their usage of both print media and Twitter. We further check if situational
updates and crisis information from Twitter is dependent on the city users
tweet from and if the corresponding use of augmenting print media indeed
offers new data not found on Twitter. Our study adds nuance by linking the
characteristics of a city, factors like GDP and literacy rate, and the bearing
these have on the discussion topics related to the second COVID wave on
Twitter in that city. Thus, our paper not only undertakes a comparative
perspective on Twitter and traditional media but augments the study through a
second order analysis of the characteristics of a city and Twitter use.

##  II Literature Review

This section will briefly discuss literature that dealt with the interplay of
COVID-19 and the information it engendered on social media platforms. During
the lockdown in India, social media usage had increased by record numbers,
[29] subsequently aiding researchers to tap into and analyse a vast pool of
data. In this section we look at past studies using social media platforms to
create situational awareness.

###  II-A COVID-19 and social media platforms

The sheer amount of online activity generated due to the COVID-19 pandemic has
prompted researchers to publish large-scale data sets across various domains
[8]. More specific data sets relating to the pandemic have been released.
Datasets of tweets specifically on vaccine hesitancy[3], panacea dataset
containing COVID’19 tweets from all over the world [2] and a multilingual
twitter data set containing tweets in different languages from India [13] are
studies that served to collect and filter data for our study. A study of
sentiment analysis of citizens via Twitter data[14] has limited their study to
India, while a study on public sentiment and opinions on tweets were collected
globally[6]. The latter also studied sentiment change over time. Twitter usage
were studied during government imposed lockdowns[1] and geospatial trends with
regards to sharing conspiracy theories and misinformation [11]. These studies
focused on key points useful to our own and helped delve deeper into our
understanding of the interplay between time, socio-economic factors, and
sentiments.

###  II-B Situational awareness, social media and media coverage

The research community has often studied trends on social media to extract
situational awareness when it comes to large-scale events such as natural
disasters [19, 24], mass emergencies [23, 26], and more recently the COVID-19
pandemic [22]. Methods to detect signals for H1N1 outbreak using crowd sourced
data [7], quantitative correlations with public health data by applying a
topic model to health-related tweets[5] were reported. A study undertook
mobility patterns before and after the WHO declared the pandemic using Twitter
data,[15] while another examined the communication between government agencies
and citizens on Twitter [16]. Causal links between media coverage and surge in
COVID-19 cases were also studied. These studies further investigated how media
coverage mitigated the spread of COVID-19 [27]. Finally, multi-modal fusion
based studies that combine various modes of information for situational
awareness have been undertaken [18, 19].
Previous studies have investigated the creation of situational awareness
explicitly using newspapers to confirm influence of mass media on public
opinions and reactions in the context of H1N1[30] and Ebola[31]. However, to
the best of our knowledge, no study has comprehensively investigated
situational awareness from both Twitter and print media, specifically for the
COVID-19 pandemic in India. In this study, we aim to cover this gap.

##  III Methodology

###  III-A Data Collection

We analyze and compare the discussions around COVID-19 during the second wave
in India on online social media platform Twitter and two offline print media
sources from India among the top 10 newspapers by readership, [16] Times of
India [9] and Hindustan Dainik.[21]

####  III-A1 Time period considered

In order to consider a representative time period during the second wave and
obtain a substantial amount of discussion on the issue, we fit a Gaussian
curve on the graph of daily new cases vs. dates where the peak of the wave
corresponds to the center of the curve. Following this, we took dates that are
within two standard deviations from the center, as the considered time period
for our study. The date range considered is from 11th April 2021 to 30th May
2021. This accounts for the buildup to the peak as well as the decline from
the peak.

####  III-A2 Twitter

We consider English language tweets from India during the mentioned time
period for our study. For collecting tweets, we first obtained unique tweet
IDs for all COVID-19-related tweets on Twitter from the Panacea dataset [2]
and processed them using the Twarc API. 111https://twarc-
project.readthedocs.io/en/latest API. To get English tweets from accounts in
India, we utilized the location field present in the tweet metadata.
Extracting via geo-coordinates is a challenging task since it is an optional
feature for users and less than 0.5% turn on their location services due to
privacy concerns [32]. Hence, we collected a mapping of Indian cities to
states and then did a fuzzy-wuzzy 222https://pypi.org/project/fuzzywuzzy/
string comparison of the location field with this mapping. Thus for every
Indian tweet, we obtained the corresponding geographic location and city.

####  III-A3 Newspapers

To collect news articles we used the online archives of the Times of India and
Hindustan Dainik. We used the beautiful soup library 333https://beautiful-
soup-4.readthedocs.io/ to scrape news from both the archives and identified
location. We further filtered news articles obtained using keywords for
COVID-19 relevant news articles. Prior studies related to extracting
situational awareness or crisis communications use keywords like “#ebola" for
Ebola virus [26] and "zika" for Zika virus [23]. Further, to include data
regarding the crisis of medical resource unavailability [28] at the time, we
included keywords relating to Covid treatments. Hence our English keywords for
filtering were - ‘corona’, ’covid’, ’covid-19’,’coronavirus’ ,’vaccine’,
’vaccination’, ’covishield’, ’covaxin’, ’oxygen’, ’plasma’, ’cylinder’,
’remdesivir’, ’ventilator’, ’bed’, ’cremation’, ’crematorium’, ’rt-pcr’,
’rtpcr’ For Hindi, we considered the corresponding Hindi translation of the
above phrases.

For our investigation, we used cities that either had the highest number of
English news articles or the highest number of tweets, or both. We selected 2
cities in each category. We also ensured that no two cities were from the same
state or proximate geographic locations, as they would face similar issues
governed by the same state government administration. Table I mentions the
cities we considered along with the number of articles and tweets in each
city. However, while considering Hindi Newspapers, we only found articles for
Delhi, Mumbai and Patna as the local language newspapers for other 3 cities in
our study are not Hindi. In the following section, we provide a qualitative
and quantitative summary of the data collected and analyzed for the study. The
goal is to understand the diversity of information present, the diversity of
topics and to investigate whether augmenting differing sources yields an
increase or decrease in diversity of information. The quantitative and
qualitative measures of information diversity are explained in later sections
of the paper. Analyzing which source of media provides the most comprehensive
information is beyond the scope of this study.

TABLE I: Number of news articles and tweets for each city City | English News | Hindi News | Tweets
---|---|---|---
Delhi | 201 | 578 | 148768
Mumbai | 192 | 122 | 105133
Hyderabad | 187 | 0 | 43459
Kokata | 132 | 0 | 19520
Ahmedabad | 201 | 0 | 7530
Patna | 254 | 352 | 5345

###  III-B Data Analysis

We employed two different modes of analytics research for the huge volume of
tweets [over 1 million] and the relatively low number of news articles [200 in
number]. We used the open coding-grounded theory approach to extract topics
from news. Prior work [26, 22] suggests that during epidemics, information
found on Twitter is either describing symptoms, discussions about prevention
or transmission, or treatment and death reports. Such information is useful
for people looking for preventive measures, those already ill, government and
health monitoring agencies. Hence we did a similar analysis of Tweets and News
content. We analyzed news articles of 6 cities and extracted sub-topics based
on the following:

  * •

“What is being talked about’ - This helps in identifying issues being debated.

  * •

“Who are the actors involved” - Here we identify key actors driving debates
around issues. We found that the second wave of COVID, mostly involved the
public , government bodies, hospitals, judicial bodies like the high court and
Supreme court, and private corporations.

For extracting topics out of tweets, we first extracted sentence embeddings
using the SBERT architecture [10]. We then used KMeans clustering, based on
the elbow method, to cluster tweets for every city. This gave us clusters as
mentioned in Table II. To analyze clustering results, we used the following
methods 1) plotting the word cloud for every cluster. 2) getting the count for
every hashtag in a cluster and choosing the top 10 hashtags and 3) For every
cluster k, we took the mean of all the embeddings present in the cluster,
${\mu}$, and then for every embedding, i present in the cluster, we computed
its cosine similarity score with ${\mu}$ denoted as ${s_{i}}$ and sorted all
the ${s_{i}}$ in decreasing order. We then manually analyzed top 50 sentences
to get a complete sense of the topics discussed in each cluster. We refer to
these topics as sub-topics.

After extracting the sub-topics out of both news articles and tweets, we
created categories to group similar sub-topics into one category. For example,
we saw different clusters and news themes for government imposing lockdown,
them starting vaccination for 18+ or starting Oxygen Express to deliver oxygen
in the critical states of India, all these sub-topics can be clubbed into one
single category of “government dealing with COVID’19 crisis". We then analyzed
every single category & sub-topics present in them, for both news and Twitter
to find out the differences. Additionally, we cross-checked sub-topics found
on Twitter that don’t occur in our news articles in TOI/Hindustan Dainik
published during the time period. Similarly, for the sub-topics found in print
news but not on Twitter we undertook a keyword search of tweets to understand
the rate and intensity of discussion among the identified sub-topics. Sub-
topics involving black-marketing of medicines, suicides committed by COVID
patients or cab drivers, and food delivery guys getting infected were not
heavily discussed in some cities on Twitter.

TABLE II: Number of clusters generated for each city City | #Clusters for Articles
---|---
Delhi | 140
Mumbai | 100
Hyderabad | 75
Kolkata | 30
Ahmedabad | 25
Patna | 18

###  III-C Quantitative Analysis

As situational updates on Twitter for a particular city will depend on the
number of people using the platform in that city, we decided to investigate
the co-relation in the diversity of information extracted from news articles
and Twitter data and a further correlation with the socio-economic
characteristics of a city. We do a quantitative study as follows: For each
city, we obtain the set of all unique keywords discussed on the English news
(N) and English twitter (T), along with the frequency of appearance for each
keyword. We pre-processed both news and tweets. For both news articles and
tweets, we removed stop words, non-ASCII characters, punctuation followed by
lemmatization and lower casing. For tweets, we removed the links, user
mentions and hashtags. Considering the high number of Twitter keywords and to
filter out scattered one-off mentions, we consider tweets with keywords that
have a minimum frequency of 10.

In order to study this quantitatively, we calculate two metrics -

  * •

`Metric 1`: The Jaccard index (intersection over union) gives us an estimate
of the overlap between news and twitter keywords.

$\frac{T\cap N}{T\cup N}$

  * •

`Metric 2`: We also define a metric to quantify news keywords left out from
twitter discussions. This is the ratio of unique news topics (only discussed
on the news) to all the news topics. A lower value would enable Twitter data
to bring out more topics discussed on the news and vice versa

$\frac{N-(T\cap N)}{N}$

###  III-D City Profiling

For cities, we consider socio-economic attributes like GDP (in $Bn), internet
penetration (IP in %) , literacy (Lit. in %), population (in millions) and the
number of COVID cases as shown in Table III. We investigate if the disparity
in situational updates on Twitter and print media could be attributed, even if
partially, to the location we are studying. We consider two cities with the
highest GDP in India i.e. Delhi and Mumbai. These coincidentally have the
highest number of Tweets (as seen in Table I), which can be attributed to
higher internet penetration. Moreover, Mumbai is the financial and business
capital of the country, while Delhi is the political capital and home to the
central government. These cities are also India’s largest metropolitan areas
[20] bearing the highest number of COVID cases in India (Delhi - 711,169) and
(Mumbai - 194,110). Ahmedabad and Patna saw fewer tweets overall but a
significant number of news articles. Ahmedabad has a much higher GDP than
Patna, the capital of one of India’s least developed states, Bihar
444https://en.wikipedia.org/wiki/List_of_Indian_states_and_union_territories_by_Human_Development_Index
& city producing the least amount of Tweets but a no dearth of news articles
both in Hindi and English. This city also has the lowest GDP and internet
penetration rate, adding to the diversity of cities we consider for our study.
Both Hyderabad and Kolkata have a comparable amount of Tweets and News
Articles. From Table III, it can be shown that Kolkata has double the
population, higher literacy and significantly higher GDP. In the next section,
we explore city wise situational updates, government action and critical
information circulation.

TABLE III: Socio economic factors of cities considered. GDP refers to Gross Domestic Product in $B, IP refers to Internet Penetration in %, Lit refers to Literacy in %, Populn refers to Population in %. City | GDP | IP | Lit. | Populn. | #Cases
---|---|---|---|---|---
Delhi | 210 | 61.068 | 88.7 | 25.83 | 711169
Mumbai | 240 | 55.820 | 84.8 | 25.97 | 194110
Kolkata | 150 | 41.216 | 80.5 | 18.54 | 153875
Hyderabad | 74 | 39.886 | 72.8 | 8.61 | 145324
Ahmedabad | 80 | 47.270 | 82.4 | 7.78 | 153711
Patna | 30 | 31.693 | 70.9 | 3.60 | 84944

##  IV Findings

As discussed in our research agenda, we found disparity in topics discussed in
the news and Twitter. We found eight major categories of missing topics which
include government organizations, MNCs/NGOs, Supreme Courts taking actions to
deal with the pandemic/crisis, hospitals giving information related to
preventive measures, effects on the economy due to lockdown, black marketing,
inadequate distribution of hospital resources and doctors involved in
malpractices. We further club them into three broad themes which are discussed
as follows.

###  IV-A Stakeholders

In this section, we will briefly discuss the response to the pandemic by
specific stakeholders of the Indian socio-economic system and the discourse
surrounding them. We will limit our discussion to the actions of the central
and state governments, the Supreme Court, Doctors, Police, and certain
organizations. We have filtered these specific stakeholders, as the majority
of the discourse found on both print media and Twitter was centered around
them. The actions taken by these stakeholders are critical for people affected
by the pandemic, hence, actions and decisions made by key stakeholders can
make a profound difference in the affected populations. In the following
subsections, we will briefly discuss the actions and decisions of these
stakeholders in the cities under study. We will further correlate the
discourse generated as a direct result of the actions of the stakeholders and
point out the differences in information found in our two comparative mediums:
print and online. Additionally, we will report certain socio-economic
markers(GDP, Internet penetration, etc) that can provide more context around
both decisions made by the specific stakeholders as well as the topics from
the emergent discourse.

####  IV-A1 Government’s handling of the crisis

The pandemic was the first of its kind, given the scale at which disrupted the
everyday of society. Governments at all levels were expectedly unprepared to
deal with the crisis. Central and state governments provided aid to citizens
in myriad ways, such as setting up oxygen cylinder plants, launching community
kitchens, and setting up medical teams to name a few. They also provided
family pensions and government jobs to family members of doctors and health
personnel who succumbed to death. While help was provided in all the cities,
discourse for the same showed a varying disparity between Twitter and print
media among the cities considered. In the city of Patna, the capital of Bihar,
tweets discussing the aid provided by the local and state government were low
in number. This is potentially because internet penetration is lower in Patna
compared to the other cities, and consequently lower number of Twitter users.
In contrast, the aforementioned aid provided in Patna was covered widely in
print media (newspapers). Cities like Mumbai and Delhi (both have high GDP)
show similar discussions on Twitter and English print media. However, Hindi
news was able to bring out topics not mentioned in both. For example, Hindi
news covered arrangements made for special trains to transport migrant workers
back to their hometowns. Hindi news media also discussed the testing
facilities setup for early detection that were instrumental for curbing the
initial rise in cases. Localised rules and reforms were rolled out in cities
to manage any crisis specific to those cities. For instance, in Kolkata, cases
rose concurrently with election time and rules regarding overcrowding at
political rallies were put in place. Discourse around the above was low on
Twitter, but covered by the print media. In some cases, we also notice a
difference in sentiments towards the government. For example - Tweets in
Mumbai reflect the oxygen shortage and people complaining about the same.
However, print media (especially Hindi news) reflects positively on the local
government appreciating the Mumbai Municipal Corporation. Officials were
praised for using oxygen tanks installed in the first wave.

####  IV-A2 Doctor’s handling of the crisis

Twitter was less widely used in cities like Ahmedabad and Patna as was
reflected in the count of total tweets sampled from these regions. As a
result, medical advice was primarily conveyed through print media in these
cities. Medical professionals used media outlets to spread awareness regarding
complications such as black fungus and methods to treat them amongst medical
staff that were operating in rural areas. Even in a city like Mumbai, which
has had a high overlap in information found on Twitter and in print media,
there was information regarding medical treatments in print media not very
prevalent in tweets. For example, information regarding Remdesivir tablets and
its efficacy on other factors such as oxygen therapy, vitamin supplements,
steroids were found in news articles, in contrast to the 4 tweets out of
hundreds of thousands in the Mumbai region. From the news articles, it appears
youth and women were ignored while providing COVID care in cities like Delhi
(the city which produced the maximum cases in India) whereas Twitter does not
mention any such scenario. Help was given by hospitals to these sets of people
by launching specialized care units. We see a similar scenario in Patna, where
the second wave was harsh for children and hence doctors took special measures
directed at the Youth by increasing beds and ICU units in the pediatric wards.
Such type of information is missing from Twitter, thus excluding an entire
demographic.

####  IV-A3 High Court/Supreme Court handling of the crisis

In almost all cities the Supreme Court and High Court intervened to take
matters into their hands and make decisions as the government faltered in
handling the crisis. In acute contexts of stress like Delhi, there are news
articles about high court directing all retailers to cap the price of medical
equipment below retail prices and also suggested the Delhi Government provide
oximeters for economically weaker sections. However there is no discussion for
these topics on Twitter. A team of medical officers were summoned to perform
audits for the supply and distribution of oxygen in various zones. This sub-
topic has 30/148768 tweets talking about the audits. Newspapers draw attention
to irregular practices across cities during the second wave: in Hyderabad,
police officials stopped ambulances coming outside Telangana district, to
avoid increasing cases in the region. In Kolkata, the high court strictly
enforced rule flouting norms during poll campaigns. However, there is no
discussion on Twitter on the above. In Ahmedabad, the high court insisted that
the state government give priority to vaccinating those wait listed for the
second dose of vaccine and questioned the availability of doses, resource
shortage, planning , management and distribution strategies.

####  IV-A4 MNCs/NGOs handling of the crisis

Print media mentions that the city of Hyderabad, covered telemedicine services
were launched by the MNCs/NGOs as an outreach strategy. Reports also mention
the work of social activists extending help to book slots for vaccine doses
and helping people register on the official CoWin
platform.555\https://www.cowin.gov.in/ (India’s vaccine registration portal)
Reports mention the police launching aid programs, websites for plasma
donation and oxygen tanks to help the resource poor. All of the above topics
do not find a mention on Twitter.

###  IV-B Resource Management

Here we discuss data around management of medical resources like oxygen
cylinders, beds and Remdesivir injections throughout the pandemic. From both
News and Twitter we observe with the rise in cases, there was a decline in
medical supplies, fostering an environment for black marketing and doctors
engaging in malpractices selling medical supplies at high prices and admitting
fake patients. In the following subsections we detail how discussions around
resource management vary between cities and the two platforms.

####  IV-B1 Repletion of hospital resources

We see cities like Ahmedabad and Patna, publicizing a large number of news
articles capturing the “why” and “how” of resource mismanagement and an
absence of mention on Twitter. Ahmedabad reported the Remdesivir crisis to the
improper distribution of injections amongst patients. Reports touched upon
medical care extended to non-critical patients leading to a shortage of supply
amongst critical patients. COVID beds and oxygen supplies were extended to
asymptomatic patients. Doctors went on strike due to the government not
responding to their demands of increasing wages thereby causing a shortage of
treatment. None of these saw mentions on the Twitter platform.

####  IV-B2 Malpractices by Doctors

This particular sub-theme is only discussed in the print media for all cities
with no mention on Twitter. In Kolkata, hospitals created an artificial
shortage of medical supplies forcing the public to buy at exorbitant prices
charged by local pharmacists. Ahmedabad and Patna were reported charging high
rates for hospital beds, oxygen cylinders beyond the government cap. Delhi
hospitals indulged in communicating false information about the availability
of beds and selling drugs at astronomical prices.

####  IV-B3 Black Marketing

In high GDP cities like Mumbai and Delhi, there is an active discussion
concerning black marketing of oxygen cylinders, Remdesivir vials, and COVID
medicines being sold at exorbitant prices on both Twitter and Newspapers.
However, in cities like Ahmedabad (GDP being comparatively lower) and Patna
(with a much lower GDP), we see little to no discussion (Ahmedabad only 2 out
of 7530 tweets reflect this and Patna showed zero tweets). A specific case in
Kolkata, oxygen cylinders were sold at exorbitant prices and arrests were
made- interestingly there is a Twitter discussion involving the black-
marketing, but none on the arrests.

###  IV-C Economic Impact of COVID

In this section, we analyze discussions around the effects that the pandemic
had on the economy. We undertake a comparison of discussion topics on twitter
and print media, the location of this data contributing towards disparity in
discussions on both mediums. In high GDP cities like Delhi and Mumbai, we
gather a similar picture about the impact on businesses and economic losses
incurred, on both News and Twitter. However, in lower GDP cities like Patna
and cities having a lower number of Tweets like Ahmedabad, we notice a
disparity in discussions about the same issue between News and Twitter.
Discussions around the government-imposed restrictions on the number of guests
allowed at weddings and social gatherings in Patna and the ensuing losses
suffered by the food and beverage industry were reported only in the print
news media. In Kolkata, newspapers reported the fear disseminating in the
airline industry about low flight loads caused by mandatory RT-PCR testing for
air travellers.Twitter offered no discussion on topics related to the airline
industry.

###  IV-D Quantitative Analysis

Differences in situational updates across cities reported in the print media
news and on social media can be attributed to key socio-economic factors among
these cities. Table IV explains the two metrics used to quantify the coverage
of news topics in Twitter discussions and estimating the overlap between
situational updates discussed on Twitter and news for every city

TABLE IV: Metrics computed City | Metric 1 | Metric 2
---|---|---
Delhi | 0.328 | 0.474
Mumbai | 0.325 | 0.488
Hyderabad | 0.322 | 0.604
Kokata | 0.250 | 0.717
Ahmedabad | 0.147 | 0.849
Patna | 0.080 | 0.919

We computed the cross-correlation of each metric with factors like GDP and
internet penetration to explore socio-economic factors influencing disparity
in discussion of keywords. From Table V, we noted strong correlations between
our metrics and socio-economic factors.

GDP: There is a high positive correlation between GDP and Metric 1. This
signifies that there is a high overlap of keywords discussed on twitter an in
the news in areas with high GDP. There is also a strong negative correlation
with Metric 2, which denotes very few news keywords that don’t reach twitter
discussions. Both these metrics denote that twitter can be used as a reliable
source for analyzing discussions in cities with higher economic prosperity,
simultaneously displaying very low values for Metric 1 and a high value for
Metric 2 (from Table V in cities with low GDP, like Patna).

Internet Penetration: There is a high positive correlation with Metric 1,
which can be attributed to more number of people having access to the
internet. There is also a strong negative correlation with Metric 2, which
denotes that there are very few news keywords that don’t reach twitter
discussions. Both these metrics denote that twitter can be used as a reliable
source for analyzing discussions in areas with high internet penetration.
Places with low internet penetration, for example Patna, reveal disparity in
discussions on Twitter and the news. In such places including offline
platforms like news in the study becomes imperative to fully capture public
discussion.

TABLE V: Correlation of Metrics with Socio-Economic Factors City | Metric 1 | Metric 2
---|---|---
Gross Domestic Product | 0.743 | -0.842
Internet Penetration | 0.659 | -0.785

Adopting qualitative and quantitative analysis helped us examine not only the
differences in the nature of information but also illumining the disparities
in the situational information discussed in different cities over the two
kinds of media. Based on the type of city, the benefits of augmenting news and
social media vary. In cities like Delhi and Mumbai, where people use the
Twitter platform most Twitter discussions overlap with what is present in the
news. However, in cities like Patna or Ahmedabad with weak Twitter use,
augmenting both types of media brings diversity to media information and
situational awareness.

##  V Discussion

From our findings we observe disparity in topics discussed on Twitter and News
across different cities. For example, certain topics discussed in the news are
either completely absent or hardly discussed on Twitter. We further see a
strong correlation between the above disparity in the news and on Twitter and
socio-economic factors [like GDP and Internet Penetration] in the cities that
are reporting news or tweeting. We found a high positive correlation between
GDP of a city and the overlap of discussion on Twitter and News. There are
several topics discussed significantly on both platforms. These range from
governments imposing lock downs, hospitals setting up COVID care centers,
state governments coming forward and providing oxygen tanks, reporting the
number of new cases/deaths and positive rates, vaccine shortage,
encouragements to get vaccinated, and political parties blaming each other for
rising cases (However, the last topic is significant on Twitter than in print
news). There is an adequate amount of news offering information that is absent
from Twitter. In resource-poor places, where internet penetration is low, we
observe that news is more representative of the lives of common people. The
news is more localized that Twitter fails to represent, especially in cities
with weaker internet penetration and lower GDP.
A striking similarity which we observed in high-income (Delhi) and low-income
(Patna) city revolves around doctors targeting resources mostly towards
affected children during the second wave. We don’t see tweets around this
occurrence. Further, we see news articles cover communities who are otherwise
missing on Twitter. For example, several news articles report the status of
blue-collar workers, the socially and marginalised like Maoists] during the
pandemic while Twitter has no mention of them. News articles report NGOs
providing help to remote areas and launching special services that aided
vaccinations. Moreover, many doctors gave treatment-related advice regarding
the COVID pandemic in newspapers rather than on Twitter. Malpractices by
hospitals creating artificial shortages in oxygen cylinders and selling at
higher prices are some of the topics mentioned only in print media
irrespective of the city.
Further, we had observed the cities of Patna, and Ahmedabad, overlap in the
number of tweets about the Delta wave and share similar discussion topics in
news articles and the Twitter platform. For instance, doctors offered COVID-
related advice in the news instead of tweeting about it, and news articles
captured how and why medical supplies get replenished and absent on the
Twitter Platform. The latter broadly reflected shortage of resources. The
above cities are otherwise socio-economically varied and our findings indeed
suggest that discourse is varied in a few cases. For example - news articles
bring out the state government focusing on inoculation of disabled people in
Ahmedabad. We don’t see such articles in Patna. Our findings suggest that for
some topics, the extent of overlap of discussions in digital and print media
indeed depends on which city is being considered. We see discussions around
the lack of oxygen cylinders and Remdesivir injections as well as a few other
illicit practices covered in print media. However, we observed engaged
discussions only in the bigger metropolitan cities of Delhi, Mumbai, and
Hyderabad on Twitter. In metros like Kolkata and Ahmadabad, the intensity of
similar discussion is not present (<0.01% of total tweets) while in Patna, a
relatively smaller city, we don’t see the above topics being tweeted. These
differences can be attributed to the economic development of the cities and
varying internet penetration. Further, we observed both English and Hindi news
sources have significant overlap in discussion topics, but Hindi news, being
more localized, is able to engage with topics that did not surface in the
English news. For example, there were incidents of commotion created at
hospitals and community kitchens by people due to lack of medical resources
and food. Another example highlighted in the Hindi news is about doctors and
front-line workers being underpaid, especially those who worked tirelessly
during the second wave to treat COVID patients.

##  VI Conclusion

In this study, we examine two types of media, print newspapers and the social
media platform, Twitter, covering the second COVID wave in India. We suggest
that a combined analysis of both helps to surface diverse situational
information in the context of the second delta wave. Extracting more
information in the domain of situational awareness significantly aids
organizations and people at large to make decisions that are in their best
interest. Moreover, it also helps the post-disease and pre-disease communities
deal with pandemics effectively by consolidating information about treatments,
symptoms, and aid among others. Our study revealed in cities where internet
penetration is low with restricted access to Twitter, print media can help
provide information filling the gap that internet-enabled platforms could
provide. We highlight that adding regional newspapers assists in enriching the
topics found in print media. Our future work would extend this study to other
regional languages like Gujarati, Marathi, and Bengali. We also aim to explore
augmenting Twitter and Newspapers with live news channels. Situational updates
given by Twitter are real-time, which is not the case for newspapers and until
the time infrastructure improves across cities in India, augmented social and
print media remain a good source of situational information.

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# Jaco-Type Graphs and Black Energy Dissipation

Johan Kok  Tshwane Metropolitan Police Department
City of Tshwane, South Africa
<EMAIL_ADDRESS>N. K. Sudev Centre for Studies in Discrete
Mathematics
Vidya Academy of Science & Technology
Thalakkottukara, Thrissur, India.
<EMAIL_ADDRESS>K. P. Chithra Naduvath Mana, Nandikkara
Thrissur, India.
<EMAIL_ADDRESS>U. Mary Department of Mathematics
Nirmala College For Women
Coimbatore, India.
<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce the notion of an energy graph as a simple,
directed and vertex labeled graph such that the arcs $(u_{i},u_{j})\notin
A(G)$ if $i>j$ for all distinct pairs $u_{i},u_{j}$ and at least one vertex
$u_{k}$ exists such that $d^{-}(u_{k})=0$. Initially, equal amount of
potential energy is allocated to certain vertices. Then, at a point of time
these vertices transform the potential energy into kinetic energy and initiate
transmission to head vertices. Upon reaching a head vertex, perfect elastic
collisions with atomic particles take place and propagate energy further.
Propagation rules apply which result in energy dissipation. This dissipated
energy is called black energy. The notion of the black arc number of a graph
is also introduced in this paper. Mainly Jaco-type graphs are considered for
the application of the new concepts.

Keywords: Energy graph, black energy, black arc number, black cloud, solid
subgraph, Jaco-type graph.

Mathematics Subject Classification: 05C07, 05C38, 05C75, 05C85.

##  1 Introduction

For general notations and concepts in graphs and digraphs see [1, 2, 3, 10].
Unless mentioned otherwise all graphs in this paper are simple, connected and
directed graphs (digraphs).

We begin by observing that any unlabeled, undirected simple and connected
graph of order $n\geq 2$ allows a vertex labeling and orientation such that
all arcs $(u_{i},u_{j})\notin A(G)$ if $i>j$. This can be established as
explained below.

Any arbitrary vertex can be labeled $u_{1}$ and all edges incident with
$u_{1}$ can be orientated as out-arcs. Thereafter, all neighbours of $u_{1}$
can randomly be labeled $u_{2},u_{3},\ldots,u_{d_{G}(u_{1})}$. Sequentially,
the resultant edges (if any) of vertex $u_{2}$ can be orientated as out-arcs
to its resultant neighbours, followed by similar orientation of the resultant
edges of $u_{3},u_{4},\ldots,u_{d_{G}(u_{1})}$. Proceeding iteratively, the
graph $G$ allows a vertex labeling and orientation as prescribed.

The above mentioned process of labeling the vertices and assigning orientation
to the edges of a given graph is indeed well-known and called topological
ordering. For the purposes of this study the digraph thus obtained is called
an energy graph.

The vertex $u_{1}$ with $d^{-}(u_{1})=0$ is called a source vertex and the
vertex $u_{n}$ and possibly others with out-degree equal to zero are called
sink vertices.

In view of the above definition on energy graphs, the following is an
immediate result.

######  Lemma 1.1.

An energy graph of order $n\geq 2$ has at least one source vertex and at least
one sink vertex.

###### Proof.

Part (i): Firstly, that some energy graphs have exactly one source vertex
follows from the topological ordering. Now, if topological ordering begins
with a number of non-adjacent vertices say, $\ell<n$ such vertices, the
vertices can randomly be labeled $u_{1},u_{2},u_{3},\ldots,u_{\ell}$.
Completing the vertex labeling and edge orientation as described clearly
results in $\ell$ source vertices. Therefore, an energy graph has at least one
source vertex.

Part (ii): In a similar way we can establish the existence of at least one
sink vertex. ∎

In general, a simple and connected digraph of order $n\geq 2$ does not
necessarily allow a vertex labeling to render it an energy graph. However, it
can be observed that a simple, connected acyclic digraph always allows such
vertex labeling.

###  1.1 Black Energy Dissipation within an Energy Graph

The energy propagation model obeys the following propagation rules. An energy
graph per se has zero mass. Each vertex $u_{i}$ is allocated a number equal to
$d^{+}(u_{i})$ of atomic particles, all of equal mass $m>0$. All the atomic
particles are initially at rest at their respective vertices.

All source vertices $u_{k}$ are initially allocated an equal amount of
potential energy of $\xi>0$ joules. The allocated potential energy is shared
equally amongst the atomic particles. At time $t=0$, all source vertices
simultaneously ignite kinetic energy, and only one atomic particle per out-arc
transits an out-arc towards the head vertex. Hence, when transition ignites
for each source vertex $u_{k}$ with $d^{+}(u_{k})=\ell$ we have
$\ell\cdot\frac{1}{2}mv^{2}=\xi$ joule.

Regardless of the true speed $v$ of an atomic particle, the time to transit
along an arc is considered to be 1 time unit. Therefore, the minimum
travelling time $t$ for some atomic particle to reach a head vertex $u_{j}$
from $u_{i}$ is equal to the directed distance between $u_{i}$ and $u_{j}$
i.e. $t=d^{\rightarrow}_{G}(u_{i},u_{j})$.

Upon one or more say, $\ell^{\prime}$ of the first atomic particles reaching a
head vertex $u_{j}$ simultaneously, the atomic particles merge into a single
atomic particle of mass $\ell^{\prime}m$ and engage in a perfect elastic
collision with the atomic particles allocated to the head vertex $u_{j}$.
Clearly these atomic particles transitted from one or more source vertices
such for any such source vertex say, $u_{i}$ the directed distance
$d^{\rightarrow}_{G}(u_{i},u_{j})$ is a minimum over all directed distances
from source vertices to $u_{j}$. For the perfect elastic collision the laws of
conservation of momentum and total energy apply. The kinetic energy of all
atomic particles reaching the head vertex $u_{j}$ later
($d^{\rightarrow}_{G}(u_{i},u_{j})$ not a minimum) than those which arrived
first, together with the mass-energy equivalence, $mc^{2}$, ($c$ is the speed
of light) dissipate into the surrounding universe. This dissipated energy is
called black energy, denoted $\mathfrak{E}_{G}$. The arcs along which these
late-coming particles transmitted are called black arcs. Hence, a black arc is
an in-arc of some vertex $u_{j}$ that does not lie on a minimum directed
distance path from any source vertex $u_{i}$.

Kinetic energy reaching a sink vertex $u_{t}$ is stored (capacitated) as
potential energy $\xi_{u_{t}}^{\prime}$ joule. By the conservation of momentum
and total energy it follows that
$\mathfrak{E}_{G}=\sum\limits_{i=1}^{n}mc^{2}\cdot
d^{+}(u_{i})+\sum\limits_{(u_{k}~{}a~{}source~{}vertex)}\xi_{u_{k}}-\sum\limits_{(u_{t}~{}a~{}sink~{}vertex)}\xi^{\prime}_{u_{t}}-$
(total dissipated mass-energy equivalence). To find closure for the
expression, the last term must be determined for an energy graph $G$.

##  2 Black Energy Dissipation within Jaco-Type Graphs

For ease of introductory analysis we consider graphs with well-defined vertex
labeling and well-defined orientation. The families of graphs which naturally
offers this research avenue are Jaco graphs and Jaco-type graphs.

The concept of linear Jaco graphs was introduced in [4] and studied initially
in [4, 5]. Further studies on these graph classes have been reported in [6, 7,
8] and following these studies some other significant papers have been
published.

In [8] it is reported that a linear Jaco graph $J_{n}(x)$ can be defined a the
graphical embodiment of a specific sequence. The introductory research (see
[8]) dealt with non-negative, non-decreasing integer sequences. This
observation has opened a wide scope for the graphical embodiment of countless
other interesting integer sequences. These graphs are broadly termed as Jaco-
type graphs and the notions of finite and infinite Jaco-type graphs are as
given below.

######  Definition 2.1.

[8] For a non-negative, non-decreasing integer sequence $\\{a_{n}\\}$, the
infinite Jaco-type graph, denoted by $J_{\infty}(\\{a_{n}\\})$, is a directed
graph with vertex set $V(J_{\infty}(\\{a_{n}\\}))=\\{u_{i}:i\in\mathbb{N}\\}$
and the arc set
$A(J_{\infty}(\\{a_{n}\\}))\subseteq\\{(u_{i},u_{j}):i,j\in\mathbb{N},i<j\\}$
such that $(u_{i},u_{j})\in A(J_{\infty}(\\{a_{n}\\}))$ if and only if
$i+a_{i}\geq j$.

######  Definition 2.2.

[8] For a non-negative, non-decreasing integer sequence $\\{a_{n}\\}$, the
finite Jaco-type Graph denoted by $J_{n}(\\{a_{n}\\})$, is the set of finite
subgraphs of the infinite Jaco-type graph
$J_{\infty}(\\{a_{n}\\});n\in\mathbb{N}$.

Jaco graphs and Jaco-type graphs can be used for modelling many theoretical
and practical problems. In this paper, we discuss some of the applications of
these types of graph classes.

######  Lemma 2.1.

Finite Jaco-type graphs are energy graphs.

###### Proof.

In view of Definition 2.1 and 2.2, it can be noted directly that a finite
Jaco-type graph has both a unique source vertex $u_{1}$ and and primary sink
vertex $u_{n}$. Furthermore, $(u_{i},u_{j})\notin A(J_{n}(\\{a_{n}\\}))$ if
$i>j$. Hence, every finite Jaco-type graph is an energy graph. ∎

###  2.1 Jaco-Type Graph for the Sequence of Natural Numbers

For purpose of notation all sequences will be labeled $s_{i}$,
$i\in\mathbb{N}$. The definition of the infinite Jaco-Type graph corresponding
to the captioned sequence can be derived from Definition 2.1. We have the
graph $J_{\infty}(s_{1})$, defined by
$V(J_{\infty}(s_{1}))=\\{u_{i}:i\in\mathbb{N}\\}$,
$A(J_{\infty}(s_{1}))\subseteq\\{(u_{i},u_{j}):i,j\in\mathbb{N},i<j\\}$ and
$(u_{i},u_{j})\in A(J_{\infty}(s_{1}))$ if and only if $2i\geq j$. Note that a
finite Jaco-Type graph $J_{n}(s_{1})$ in this family is obtained from
$J_{\infty}(s_{1})$ by lobbing off all vertices $u_{k}$ (with incident arcs)
$\forall\,k>n$.

The arrival times of the atomic particles at a head vertex $u_{j}$ will be
stringed and denoted $u_{j}\sim\langle
t_{1},t_{2},t_{3},\ldots,t_{d^{-}_{G}(u_{j})}\rangle$ such that $t_{i}\leq
t_{l}$ for $1\leq i,l\leq{d^{-}_{G}(u_{j})}$. For the Jaco-type graph
$J_{8}(s_{1})$ we find (see Figure 1): $u_{1}\sim\langle 0\rangle$,
$u_{2}\sim\langle 1\rangle$, $u_{3}\sim\langle 2\rangle$, $u_{4}\sim\langle
2,3\rangle$ hence, ($\frac{1}{6}\xi+mc^{2}$) joules transiting along arc
$(u_{3},u_{4})$ dissipate as black energy. Then we have $u_{5}\sim\langle
3,3\rangle$ so two atomic particles merge to collide perfectly at $u_{5}$ with
$\frac{7}{24}\xi$ joules. Then $u_{6}\sim\langle 3,3,4\rangle$ hence,
$\frac{7}{24}\xi$ joules collide perfectly at $u_{6}$ and
($\frac{7}{72}\xi+mc^{2}$) joules dissipate as black energy. At vertices
$u_{7}$ and $u_{8}$ the arrival times are stringed as $u_{7}\sim\langle
3,4,4\rangle$ and $u_{8}\sim\langle 3,4,4,4\rangle$ respectively. At $u_{7}$
and $u_{8}$ energy amounting to $(\frac{7}{72}\xi+\frac{7}{48}\xi+2mc^{2})$
joules and $(\frac{7}{72}\xi+\frac{7}{48}\xi+\frac{1}{8}\xi+3mc^{2})$ joules
dissipate as black energy, respectively. Only $(\frac{1}{8}\xi+mc^{2})$ joules
reach the sink vertex, $u_{8}$. The energy graph has a total of
$(\frac{1}{8}\xi+9mc^{2})$ joules potential energy and mass-energy
equivalence, capacitated within the graph. Hence, a total of
$(\frac{7}{8}\xi+7mc^{2})$ joules dissipated as black energy. Figure 1 depicts
$J_{8}(s_{1})$.

Figure 1: $J_{8}(s_{1})$.

###  2.2 Jaco-Type Graph for the Fibonacci Sequence

The definition of the infinite Jaco-Type graph corresponding to the Fibonacci
sequence, $s_{2}=\\{f_{n}\\}$, $f_{0}=0,f_{1}=1,f_{2}=1$,
$f_{n}=f_{n-1}+f_{n-2}$, $n=1,2,3,\ldots$ can be derived from Definition 2.1.
We have the graph $J_{\infty}(s_{2})$, defined by
$V(J_{\infty}(s_{2}))=\\{u_{i}:i\in\mathbb{N}\\}$,
$A(J_{\infty}(s_{2}))\subseteq\\{(u_{i},u_{j}):i,j\in\mathbb{N},i<j\\}$ and
$(u_{i},u_{j})\in A(J_{\infty}(s_{2}))$ if and only if $i+f_{i}\geq j$.

For the Jaco-type graph $J_{12}(s_{2})$ we find (see Figure 2)
$v_{1}\sim\langle 0\rangle$, $u_{2}\sim\langle 1\rangle$, $u_{3}\sim\langle
2\rangle$, $u_{4}\sim\langle 3\rangle$ and $u_{5}\sim\langle
3,4\rangle,u_{6}\sim\langle 4,4\rangle,u_{7}\sim\langle
4,4,5\rangle,u_{8}\sim\langle 4,5,5\rangle,u_{9}\sim\langle
4,5,5,5\rangle,u_{10}\sim\langle 4,5,5,5,5\rangle,u_{11}\sim\langle
5,5,5,5,5\rangle$ and $u_{12}\sim\langle 5,5,5,5,5,6\rangle$. It implies that
only $(\frac{163}{450}\xi+5mc^{2})$ joules kinetic energy capacitate at the
sink vertex $u_{12}$. The energy graph has a total of
$(\frac{163}{450}\xi+21mc^{2})$ joules potential energy and mass-energy
equivalence, allocated within the graph. This means that
$(\frac{287}{450}\xi+12mc^{2})$ joules dissipate into black energy. Figure 2
depicts $J_{12}(s_{2})$.

Figure 2: $J_{12}(s_{2})$.

###  2.3 Number of Black Arcs of Certain Graphs

For a simple connected graph $G$ in general, the number of black arcs
dependent on the orientation of a corresponding energy graph. In general an
energy graph for a given graph $G$ is not unique. Denote an orientation of a
graph by $\varphi(G)$. Denote the number of black arcs in respect of a given
$\varphi(G)$ to be $b^{\varphi(G)}(G)$. Define the black arc number,
$b^{\bullet}(G)=\min\\{b^{\varphi(G)}(G):\forall\,\ \varphi(G)\\}$. First we
present a perhaps obvious, but useful lemma.

######  Lemma 2.2.

If the stringed arrival times of atomic particles at vertex $u_{j}$ in an
energy graph $G$ is $u_{j}\sim\langle
t_{1},t_{2},t_{3},\ldots,t_{\ell},t_{\ell+1},t_{\ell+2},\ldots,t_{d^{-}_{G}(u_{j})}\rangle$
and $t_{1}=t_{2}=t_{3}=\cdots=t_{\ell}<t_{\ell+1}\leq t_{\ell+2}\leq\cdots\leq
v_{d^{-}_{G}(u_{j})}$, then $d^{-}_{G}(u_{j})-\ell$ in-arcs of $u_{j}$
dissipate black energy.

###### Proof.

Because $t_{1}=t_{2}=t_{3}=\cdots=t_{\ell}<t_{\ell+1}\leq
t_{\ell+2}\leq\cdots\leq u_{d^{-}_{G}(u_{j})}$, the atomic particles at
vertices $u_{i}$, $1\leq i\leq\ell$ simultaneously arrive first at vertex
$u_{j}$ and they merge to collide perfectly with the atomic particles
allocated to $u_{j}$. In terms of the rules of the energy propagation model
all other atomic particles along the other in-arcs of $u_{j}$ dissipate black
energy. ∎

To illustrate the use of Lemma 2.2, it is now applied to star graphs, paths
and cycles. First, an obvious corollary.

######  Corollary 2.3.

$b^{\bullet}(G)=0\Leftrightarrow\mathfrak{E}(G)=0$.

Note that for an energy graph $G$ the portion of black energy resulting from
atomic particles dissipating their mass-energy equivalence is given by
$b^{\bullet}(G)\cdot mc^{2}$ joules.

######  Proposition 2.4.

For star graphs $S_{1,n}$, $n\geq 3$; for paths $P_{n}$, $n\geq 2$ and for
cycles $C_{n}$, $n\geq 3$ we have

  1. (i)

$b^{\bullet}(S_{1,n})=0$ and
$\mathfrak{E}(S_{1,n})=0,\forall\,\varphi(S_{1,n})$,

  2. (ii)

$b^{\bullet}(P_{n})=0$ and $\mathfrak{E}(P_{n})=0$.

  3. (iii)

$b^{\bullet}(C_{n})=\begin{cases}0,&n\ \text{is even},\\\ 1,&n\ \text{is
odd},\end{cases}$
and $\mathfrak{E}(C_{n})=\begin{cases}0,&n\ \text{is even},\\\
\frac{1}{2}\xi+mc^{2},&n\ \text{is odd}.\end{cases}$

###### Proof.

Part (i): Consider a star graph $S_{1,n},\ n\geq 3$ and first orientate all
edges as out-arcs from the central vertex. Label the central vertex as $u_{1}$
and randomly vertex label the pendant vertices $u_{i},2\leq i\leq n+1$.
Clearly, at $t=1$ a single atomic particle has transited an arc to reach a
corresponding pendant sink vertex. All arcs were transited simultaneously and
hence $u_{i}\sim\langle 1\rangle,\ 2\leq i\leq n+1$. Hence,
$b^{\bullet}(S_{1,n})=0$. Hence, $\mathfrak{E}(S_{1,n})=0$. Now take the
inverse orientation and the reasoning remains the same.

Finally, and without loss of generality, consider the first orientation of the
star graph and inverse the orientation of any number $\ell<n$ arcs. Now the
star digraph has $\ell$ source vertices. At $t=1$, $\ell$ atomic particles
will merge at $u_{1}$ for a perfect collision to transfer $\ell\cdot\xi$
joules to the $n-\ell$ atomic points allocated to $u_{1}$. So
$u_{1}\sim\langle\underbrace{1,1,1,\ldots,1}_{\ell-entries}\rangle$. The
aforesaid kinetic energy will equally divide amongst $n-\ell$ atomic
particles, each transited an arc to reach a corresponding pendant sink vertex,
$u_{j}$. For each sink vertex we have, $v_{j}\sim\langle 2\rangle$. Total
energy is conserved hence, $\mathfrak{E}(S_{1,n})=0$ implying
$b^{\bullet}(S_{1,n})=0$. The result holds for all $\varphi(S_{1,n}),\ n\geq
3$.

Part (ii): Consider any path $P_{n}$, $n\geq 2$ and label the vertices to have
the path $u_{1}u_{2}u_{3}\ldots u_{n}$ and orientate with arcs
$(u_{i},u_{j})$, $1\leq i\leq n-1$ and $j=i+1$. Clearly, on igniting
propagation of energy and atomic particle mass, we have $u_{i}\sim\langle
i-1\rangle$, $1\leq i\leq n$. Since a single atomic particle arrives at each
vertex $u_{j}$, $j\neq 1$ no black edges exist. Therefore,
$b^{\bullet}(P_{n})=0\Rightarrow\mathfrak{E}(P_{n})=0$.

Part (iii)(a): Consider a cycle $C_{n}$, $n\geq 4$, $n$ even. Let the vertices
of $C_{n}$ be located along the circumference of a circle with vertex $u_{1}$
at the exact-top, and $u_{n}$ at the exact-bottom of the circle. From $u_{1}$
label the vertices located anticlockwise consecutively,
$u_{2},u_{3},u_{4},\ldots,u_{\frac{n}{2}}$ and orientate all corresponding
edges including $u_{\frac{n}{2}}u_{n}$, anticlockwise. Similarly, from $u_{1}$
label vertices located clockwise consecutively,
$u_{\frac{n}{2}+1},u_{\frac{n}{2}+2},\ldots,u_{n-1}$ and orientate all
corresponding edges including $u_{n-1}u_{n}$, clockwise. Clearly, this
construction of an energy cycle does not contradict generality. Hence, on
igniting propagation of energy and atomic particle mass we have
$u_{i}\sim\langle i-1\rangle$, $1\leq i\leq\frac{n}{2}$ and
$u_{\frac{n}{2}+i}\sim\langle i\rangle$, $1\leq i\leq\frac{n}{2}-1$.
Therefore, $u_{n}\sim\langle\frac{n}{2},\frac{n}{2}\rangle$. In view of Lemma
2.2, no black arc exists. Therefore,
$b^{\bullet}(C_{n})=0\Rightarrow\mathfrak{E}(C_{n})=0$ if $n$ even.

Part (iii)(b): The vertex labeling and orientation of a cycle $C_{n}$, where
$n\geq 3$ and is odd, follow similar to Part iii(a) with the anticlockwise
vertex labels running through
$u_{2},u_{3},\ldots,u_{\lfloor\frac{n}{2}\rfloor}$ and the clockwise vertex
labels running through
$u_{\lfloor\frac{n}{2}\rfloor+1},u_{\lfloor\frac{n}{2}\rfloor+2},\ldots,u_{n-1}$.
Note that the arc $(u_{n-1},u_{n})$ results in
$u_{n}\sim\langle\lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil\rangle$. So
exactly one arc results in a black arc with black energy,
$(\frac{1}{2}\xi+mc^{2})$ joules. Therefore,
$b^{\bullet}(C_{n})=1\Rightarrow\mathfrak{E}=\frac{1}{2}\xi+mc^{2}$ because
the initial amount of potential energy capacitated at $u_{1}$ is $\xi$ joules.
∎

######  Theorem 2.5.

An acyclic graph $G$ of order $n\geq 2$ has $b^{\bullet}(G)=0$ and therefore,
$\mathfrak{E}(G)=0$.

###### Proof.

An acyclic graph $G$ is a bipartite graph. Partition $V(G)$ into disjoint
subsets $X,Y$ and without loss of generality, orientate each edge $uw$, $u\in
X$, $w\in Y$ to be the arc $(u,w)$. Label the vertices in $X$ as
$u_{1},u_{2},u_{3},\ldots,u_{|X|}$ and those in $Y$ as
$w_{|X|+1},w_{|X|+2},w_{|X|+3},\ldots,w_{|X|+|Y|}$. Clearly, the resultant
vertex labeled digraph $G$ is an energy graph. Furthermore, at $t=1$ all
atomic particles transit along an arc from a corresponding source vertex in
$X$ to a sink vertex in $Y$. Therefore, $b^{\bullet}(G)=0$. It immediately
follows that $\mathfrak{E}(G)=0$. ∎

######  Corollary 2.6.

A simple connected graph $G$ of order $n\geq 2$ which has no odd cycle has
$b^{\bullet}(G)=0$ and therefore, $\mathfrak{E}(G)=0$.

###### Proof.

It is well known that a graph which has no odd cycle is bipartite. The result
then follows from the proof of Theorem 2.4. ∎

An energy graph $G$ with $\mathfrak{E}(G)=0$, ($b^{\bullet}(G)=0$) is said to
satisfy the law of conservation of total energy. Characterisation of these
graphs follows in the next result.

######  Theorem 2.7.

An energy graph $G$ of order $n\geq 2$ satisfies the law of conservation of
total energy if and only if $G$ has no odd cycle.

###### Proof.

Case 1: If $G$ has no odd cycle the result follows from Corollary 2.6.

Case 2: Assume $G$ satisfies the law of conservation of total energy and has
an odd cycle. Hence, $\mathfrak{E}(G)=0$. However, since $m>0$ and $\xi>0$ and
order $n\in\mathbb{N}$ is finite, a vertex of the odd cycle receives or
propagates some energy $>0$ joules from in-arcs or along out-arcs,
respectively. From Proposition 2.4(iii) it follows that for all arcs along the
odd cycle, at least one such arc $(u_{i},u_{j})$ is a black arc in respect of
the odd cycle as an induced subgraph of $G$. Thus, at least
$(\frac{1}{2}mc^{2}+\epsilon)$ joules dissipate as black energy along the arc
$(u_{i},u_{j})$ during exhaustive propagation and $b^{\bullet}(G)>1$. The
aforesaid is a contradiction to the assumption. Therefore, the result holds. ∎

The Jaco-type graphs $J_{n}(s_{1})$, $J_{n}(s_{2})$ clearly have well-defined
black arcs and therefore it is possible to determine the number of black arcs.
The black arc number can be determined through the Jaco-type Black Arc
Algorithm which is presented next.

######  Definition 2.3.

In a finite Jaco-type graph $G$ of order $n\geq 2$ and a vertex $u_{j}\in
V(G)$ the out-open neighbourhood is the set of head vertices of $u_{j}$,
denoted $\mathcal{H}(u_{j})$. The set of vertices of the induced subgraph
$\langle\mathcal{H}(u_{j})\rangle$ is called the black cloud of $u_{j}$.

###  2.4 Jaco-type Black Arc Algorithm

Note that the algorithm is dense and requires iterative refinement for ICT
application.

Step 1:- For a finite Jaco-type graph $G$, set $i=1$,
$\mathcal{B}_{0}(G)=\emptyset$ and let $G_{i}=G$. Go to Step 2.

Step 2:- Set $j=i$ and consider vertex $u_{j}\in V(G_{j})$ and determine
$\langle\mathcal{H}(u_{j})\rangle$. Go to Step 3.

Step 3:- Let $\mathcal{B}_{j}(G)=\mathcal{B}_{j-1}(G)\cup
E(\langle\mathcal{H}(u_{j})\rangle)$. If $j=n-1$, go to Step 4. Else, let
$i=j+1$ and $G_{i}=G_{j}-E(\langle\mathcal{H}(u_{j})\rangle)$. Go to Step 2.

Step 4:- Set $b^{\bullet}(G)=|\mathcal{B}_{n-1}(G)|$ and exit.

We call the set $\mathcal{B}_{n-1}(G)$ the black cloud of the graph $G$. Note
that
$b^{\bullet}(G)=|\mathcal{B}_{n-1}(G)|\leq|\bigcup\limits_{j=1}^{n}E(\langle\mathcal{H}(u_{j})\rangle)|$.

######  Theorem 2.8.

For a finite Jaco-type graph $G$ the Jaco-type Black Arc Algorithm is well-
defined and it converges.

###### Proof.

Clearly, Step 1 is unambiguous and finite. Clearly, Step 2 is unambiguous and
since $G$ is finite the range $j\leq n-1$ in Step 3 is well-defined and finite
which implies Step 2 converges. Furthermore, since the out-neighbourhood of
any vertex is well-defined and arcs are unambiguous in a simple digraph, the
induced subgraph $\langle\mathcal{H}(u_{j})\rangle$ is well-defined. Since
$G_{j}$ is finite, determining $\langle\mathcal{H}(u_{j})\rangle$ converges.
Therefore, Step 3 converges. Step 4 is unambiguous and finite. Hence, the
Jaco-type Black Arc Algorithm is well-defined and it converges. ∎

######  Theorem 2.9.

The Jaco-type Black Arc Algorithm determines all the black arcs of a finite
Jaco-type graph $G$ of order $n\geq 2$.

###### Proof.

At any time $t_{i}$ which corresponds with the $i^{th}$-iteration the vertex
$u_{i}\in V(G_{i})$ is under consideration. If any two out-neighbours of
$u_{i}$ say vertices $u_{k},u_{\ell}$, $k<\ell$ are adjacent the arc
$(u_{k},u_{\ell})$ exists. No energy transmission will be possible from
$u_{k}$ to $u_{\ell}$ at time $t_{i+1}$ hence, all energy allocated at vertex
$u_{k}$ will dissipate into black energy at $t_{i+1}$. Therefore, the Jaco-
type Black Arc Algorithm determines all black arcs of $G$. ∎

######  Application 2.4.1.

Applying the Jaco-type Black Arc Algorithm on $J_{8}(s_{1})$ and
$J_{12}(s_{2})$, we have

$\displaystyle\mathcal{B}_{7}(J_{8}(s_{1}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{5},u_{6}),(u_{5},u_{7}),(u_{5},u_{8}),(u_{6},u_{7}),(u_{6},u_{8}),(u_{7},u_{8})\\},$
$\displaystyle\mathcal{B}_{11}(J_{12}(s_{2}))$ $\displaystyle=$
$\displaystyle\\{(u_{4},u_{5}),(u_{6},u_{7}),(u_{6},u_{8}),(u_{6},u_{9}),(u_{6},u_{10}),(u_{7},u_{8}),(u_{7},u_{9}),$
$\displaystyle(u_{7},u_{10}),(u_{8},u_{9}),(u_{8},u_{10}),(u_{9},u_{10}),(u_{11},u_{12})\\}$

Therefore, $b^{\bullet}(J_{8}(s_{1}))=7$, $b^{\bullet}(J_{12}(s_{2}))=12$.

The notion of primitive holes in graphs has been introduced in [9] and defined
the primitive degree, $d^{p}_{G}(u)$ of a vertex $u\in V(G)$. Let the graphs
$G_{i}$, $1\leq i\leq n-1$ be those resulting from the Jaco-type Black Arc
Algorithm. Denote the underlying graph of $G$ by $G^{*}$. The next theorem
presents $b^{\bullet}(G)$ in terms of primitive degrees found in Jaco-type
graphs.

######  Theorem 2.10.

For a finite Jaco-type graph $G$ of order $n\geq 2$ we have
$b^{\bullet}(G)=\sum\limits_{i=1}^{n-1}h^{p}_{G^{*}_{i}}(u_{i})$.

###### Proof.

Lemma 2.1 states that a Jaco-type graph is an energy graph. From definitions
2.1 and 2.2 it follow directly that each arc of the black cloud
$\langle\mathcal{H}(u_{i})\rangle$ is an edge of a primitive hole of
$G^{*}_{i}$ with the common vertex $u_{i}$. So the number of black edges
associated with $u_{i}\in V(G^{*}_{i})$ equals $d^{P}_{G^{*}}(u_{i})$. Hence,
the result $b^{\bullet}(G)=\sum\limits_{i=1}^{n-1}h^{p}_{G^{*}_{i}}(u_{i})$ is
settled. ∎

The subgraph
$G-\bigcup\limits_{i=1}^{n-1}E(\langle\mathcal{H}(u_{i})\rangle)=G-\mathcal{B}_{n-1}(G)$
is called the solid subgraph of the propagating graph $G$, denoted $G^{s}$.
For a given energy graph $G$ (given orientation), the corresponding solid
subgraph is unique. We now have a useful lemma, with trivial proof.

######  Lemma 2.11.

The number of arcs, $|\mathcal{B}_{n-1}(G)|$ equals the number of atomic
particles which dissipated into mass-energy equivalence when propagation
exhausted. Similarly the number of arcs, $|A(G^{s})|$ equals the number of
atomic particles which remained within the energy graph when propagation
exhausted.

######  Application 2.4.2.

For the Jaco-type graph $J_{8}(s_{1})$, the number of atomic particles
remained is
$|A(J_{8}(s_{1}))|-b^{\bullet}(J_{8}(s_{1}))=16-7=9=|A(J^{s}_{8}(s_{1}))|$
only and the number of atomic particles dissipated is $7$. In $J_{12}(s_{2})$,
only
$|A(J_{12}(s_{2}))|-b^{\bullet}(J_{12}(s_{2}))=33-12=21=|A(J^{s}_{12}(s_{2}))|$
atomic particles remained and $12$ dissipated.

##  3 Application to Binary Code, Gray Code and Modular Arithmetic Jaco-type
Graphs

For a binary code of bit width $n$, the the binary combinations are always
finite and an even number of combinations (that is, $2^{n}$ numbers) exist.
The binary combinations will be tabled such that consecutive rows maps to
consecutive decimal representations $b_{i}=i-1$, where $i=1,2,3,\ldots,2^{n}$
of the binary code.

Table 1 serves as an example of the convention for a binary code of bit width
$3$.

0 | 0 | 0 | $\mapsto 0=b_{1}$
---|---|---|---
0 | 0 | 1 | $\mapsto 1=b_{2}$
0 | 1 | 0 | $\mapsto 2=b_{3}$
0 | 1 | 1 | $\mapsto 3=b_{4}$
1 | 0 | 0 | $\mapsto 4=b_{5}$
1 | 0 | 1 | $\mapsto 5=b_{6}$
1 | 1 | 0 | $\mapsto 6=b_{7}$
1 | 1 | 1 | $\mapsto 7=b_{8}$
Table 1:

Contrary to the general finite Jaco-type graphs $J_{n}(\\{a_{n}\\}),\
n\in\mathbb{N}$ the graphical embodiment of the binary codes denoted,
$\mathcal{G}_{n}$ for $n\in\mathbb{N}$ is defined finite on exactly $2^{n+1}$
vertices,
$u_{1},u_{2},u_{3},\ldots,u_{2^{n}},u_{2^{n}+1},u_{2^{n}+2},\ldots,u_{2^{n+1}}$.
We define $d^{+}(u_{i})=b_{i}+1$, $1\leq i\leq 2^{n}$ and
$d^{+}(u_{j})=2^{n}-k$, $j=2^{n}+k$, $k=1,2,3,\ldots,2^{n}$.

It follows easily that $\mathcal{G}_{n}\simeq J_{2^{n+1}}(s_{1})$. See Figure
1 for $\mathcal{G}_{2}\simeq J_{8}(s_{1})$. The disconnected graph obtained
from the union of $t\geq 2$ copies of $\mathcal{G}_{n}$ has vertices labeled
such that the corresponding vertices in the $i^{th}$-copy, $i\geq 2$ are
$u_{(i-1)2^{n+1}+j}$, $2\leq i\leq t$, $1\leq j\leq 2^{n+1}$. Through easy
sequential counting and indexing known results for the finite Jaco-type graph
$J_{2^{n}+1}(s_{1})$ can be extrapolated. For example, for
$\bigcup\limits_{t-copies}\mathcal{G}_{n}$, $\mathcal{G}_{n}$ the graphical
embodiment of the binary code with bit width $n$ we have
$\mathbb{J}(\bigcup\limits_{t-copies}\mathcal{G}_{n})=\\{u_{p+j}:u_{p}\in\mathbb{J}(J_{2^{n+1}}(s_{1})),j=0,1,2,\ldots,(t-1)2^{n+1}\\}$;
$|\mathbb{J}(\bigcup\limits_{t-copies}\mathcal{G}_{n})|=t\cdot|\mathbb{J}(J_{2^{n+1}}(s_{1}))|$
and $b^{\bullet}(\bigcup\limits_{t-copies}\mathcal{G}_{n})=t\cdot
b^{\bullet}(J_{2^{n+1}}(s_{1})$.

The analysis for Gray codes follows similarly to that for binary codes since
the Gray combinations can be mapped onto exactly the same decimal
representations.

###  3.1 Jaco-type Graph for Sequences Modulo $k$

It is well known that for $\mathbb{N}_{0}$ and $n,k\in\mathbb{N}$, $k\geq 2$
modular arithmetic allows an integer mapping in respect of mod $k$ as follows:
$0\mapsto 0=m_{0}$, $1\mapsto 1=m_{1}$, $2\mapsto 2=m_{2},\ldots,k-1\mapsto
k-1=m_{k-1}$, $k\mapsto 0=m_{k}$, $k+1\mapsto 1=m_{k+1},\ldots$ Let
$s_{3}=\\{a_{n}\\}$, $a_{n}\equiv n($mod $k)=m_{n}$. Consider the infinite
root-graph $J_{\infty}(s_{3})$ and define $d^{+}(u_{i})=m_{i}$, for
$i=1,2,3,\ldots$.

From the aforesaid definition it follows that the case $k=1$ will result in a
null (edgeless) Jaco-type graph $\forall\,n\in\mathbb{N}$. For the case $k=2$
the Jaco-type graph for $n\geq 2$ and even, is the union of $\frac{n}{2}$
copies of directed $P_{2}$. For the case $k=3$ the Jaco-type graph is a
directed tree hence, an acyclic graph $G$ and therefore $b^{\bullet}(G)=0$.
The smallest mod $k$ Jaco-type graph which has an black arc is $J_{4}(s_{3})$,
$k=4$.

Figure 3 depicts $J_{12}(s_{3})$ for $k=5$.

Figure 3: $J_{12}(s_{3}),\ k=5$

We note that a Jaco-type graph $J_{n}(s_{3})$ has $\lceil\frac{n}{k}\rceil$
sink vertices. The sink vertices resulting from the modular function is called
mod-sink vertices. It follows from the Jaco-type Black Arc Algorithm that
$b^{\bullet}(J_{12}(s_{3}))=5$. A total of $(\frac{7}{24}\xi+2mc^{2})$ joules,
$(\frac{1}{24}\xi+mc^{2})$ joules and $(\frac{1}{12}\xi+2mc^{2})$ joules reach
the sink vertices $v_{5},v_{10},v_{12}$ respectively. A total of
$(\frac{47}{72}\xi+6mc^{2})$ joules dissipate as black energy along black
arcs. Therefore, a total of $(\frac{1}{12}\xi+14mc^{2})$ joules are
capacitated within the graph.

######  Lemma 3.1.

A mod-sink vertex $u_{j}$ of the Jaco-type graphs $J_{n}(s_{3})$, $k\geq 3$
has $d^{-}(u_{j})=\lfloor\frac{k}{2}\rfloor$.

###### Proof.

Consider any mod-sink vertex $u_{j}$ of the Jaco-type graph $J_{n}(s_{3})$,
$n\geq k\geq 4$. Clearly, the minimum $t$ for which the arc $(u_{t},u_{j})$
exists is $\lfloor\frac{n}{2}\rfloor$. Hence, the result. ∎

For the case $k=4$, $4\leq n\leq 35$: Applying the Jaco-type Black Arc
Algorithm iteratively leads to the next result.

######  Proposition 3.2.

For $s_{3}=\\{a_{n}\\}$, $a_{n}\equiv n($mod $4)$ the Jaco-type graph
$J_{n}(s_{3})$ has:
(i) $b^{\bullet}(J_{n}(s_{3}))=\lfloor\frac{n}{2}\rfloor-1$, alternatively;
(ii)

$b^{\bullet}(J_{n}(s_{3}))=\begin{cases}i,&\text{$\forall\,n=3+2i$,
$i=0,1,2,\ldots$ }and,\\\ i+1,&\text{$\forall\,n>max\\{\ell:\ell<n$ and
$\ell=3+2i$, $i=0,1,2,\ldots\\}$}.\end{cases}$

###### Proof.

Part (i): It is trivially true that $b^{\bullet}(J_{n}(s_{3}))\leq
b^{\bullet}(J_{n+1}(s_{3}))$. Applying the Jaco-type Black Arc Algorithm to
both $J_{4}(s_{3}),J_{5}(s_{3})$ results in the only black arc,
$(u_{3},u_{4})$ for both thus,
$b^{\bullet}(J_{4}(s_{3}))=b^{\bullet}(J_{5}(s_{3}))=1=\lfloor\frac{4}{2}\rfloor-1=\lfloor\frac{5}{2}\rfloor-1$.
Thus the result holds for the integer pair, $n=4,5$. Assume the results holds
for the integer pair $n=t,t+1$, $t\in\mathbb{N}$. Hence,
$b^{\bullet}(J_{t}(s_{3}))=\lfloor\frac{t}{2}\rfloor-1=\lfloor\frac{t+1}{2}\rfloor-1=b^{\bullet}(J_{t+1}(s_{3}))$.

Observe that if $t$ is even then
$\lfloor\frac{t}{2}\rfloor-1=\lfloor\frac{t+1}{2}\rfloor-1$. Since for,
$t^{\prime\prime}=t+2$ which could have been the initial even integer
$\in\mathbb{N}$ by the induction assumption, we have that
$\lfloor\frac{t^{\prime\prime}}{2}\rfloor-1=(\lfloor\frac{t}{2}\rfloor+1)-1\implies
b^{\bullet}(J_{t^{\prime\prime}}(s_{3}))=b^{\bullet}(J_{t}(s_{3}))+1=b^{\bullet}(J_{t+1}(s_{3}))+1=\lfloor\frac{t^{\prime\prime}+1}{2}\rfloor\rfloor-1=b^{\bullet}(J_{t^{\prime\prime}+1}(s_{3}))$.
Hence the result holds for the integer pair $t^{\prime\prime}$ and
$t^{\prime\prime}+1$ or put another way, it holds for the pair of Jaco-type
graphs, $J_{t^{\prime\prime}}(s_{3}),J_{t^{\prime\prime}+1}(s_{3})$ that
$b^{\bullet}(J_{t^{\prime\prime}}(s_{3}))=\lfloor\frac{t^{\prime\prime}}{2}\rfloor-1=\lfloor\frac{t^{\prime\prime}+1}{2}\rfloor-1=b^{\bullet}(J_{t^{\prime\prime}+1}(s_{3}))$.
The general result $\forall\,n\geq 4$, $n\in\mathbb{N}$ follows by
mathematical induction.

Part (ii): Alternatively:

Part(ii)(a): $\forall\,n=3+2i$, $i=0,1,2\ldots$ it follows from Case (i) that
$\lfloor\frac{3+2i}{2}\rfloor-1=\lfloor\frac{3}{2}+\frac{2i}{2}\rfloor-1=i$.

Part (ii)(b): Since $3+2i$, $i=0,1,2\ldots$ is odd, the integer
$n>max\\{\ell:\ell<n$ and $\ell=3+2i$, $i=0,1,2,\ldots\\}\Rightarrow n=\ell+1$
and even. Therefore, from Case (i) it follows that
$b^{\bullet}(J_{n}(s_{3}))=i+1$. ∎

If $k=5$, applying the Jaco-type Black Arc Algorithm iteratively for $n\geq
4$, we get the following table.

$n$ | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14
---|---|---|---|---|---|---|---|---|---|---|---
$b^{\bullet}(J_{n}(s_{3}))$ | 1 | 1 | 1 | 2 | 3 | 3 | 4 | 5 | 5 | 6 | 6
$n$ | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25
$b^{\bullet}(J_{n}(s_{3}))$ | 7 | 7 | 8 | 8 | 9 | 10 | 11 | 11 | 12 | 12 | 13
$n$ | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 |
$b^{\bullet}(J_{n}(s_{3}))$ | 14 | 14 | 15 | 15 | 16 | 16 | 17 | 18 | 18 | 19 |
Table 2:

The black clouds are

$\displaystyle\mathcal{B}_{3}(J_{4}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4})\\},$
$\displaystyle\mathcal{B}_{4}(J_{5}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4})\\},$
$\displaystyle\mathcal{B}_{5}(J_{6}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4})\\},$
$\displaystyle\mathcal{B}_{6}(J_{7}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7})\\},$
$\displaystyle\mathcal{B}_{7}(J_{8}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8})\\},$
$\displaystyle\mathcal{B}_{8}(J_{9}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8})\\},$
$\displaystyle\mathcal{B}_{9}(J_{10}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10})\\},$
$\displaystyle\mathcal{B}_{10}(J_{11}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11})\\},$
$\displaystyle\mathcal{B}_{11}(J_{12}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11})\\},$
$\displaystyle\mathcal{B}_{12}(J_{13}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13})\\},$
$\displaystyle\mathcal{B}_{13}(J_{14}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13})\\},$
$\displaystyle\mathcal{B}_{14}(J_{15}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15})\\},$
$\displaystyle\mathcal{B}_{15}(J_{16}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16})\\},$
$\displaystyle\mathcal{B}_{16}(J_{17}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16})\\},$
$\displaystyle\mathcal{B}_{17}(J_{18}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18})\\},$
$\displaystyle\mathcal{B}_{18}(J_{19}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18})\\},$
$\displaystyle\mathcal{B}_{19}(J_{20}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20})\\},$
$\displaystyle\mathcal{B}_{20}(J_{21}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21})\\},$
$\displaystyle\mathcal{B}_{21}(J_{22}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21})\\},$
$\displaystyle\mathcal{B}_{22}(J_{23}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23})\\},$
$\displaystyle\mathcal{B}_{23}(J_{24}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23})\\},$
$\displaystyle\mathcal{B}_{24}(J_{25}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25})\\},$
$\displaystyle\mathcal{B}_{25}(J_{26}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26})\\},$
$\displaystyle\mathcal{B}_{26}(J_{27}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26})\\},$
$\displaystyle\mathcal{B}_{27}(J_{28}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28})\\},$
$\displaystyle\mathcal{B}_{28}(J_{29}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28})\\}$
$\displaystyle\mathcal{B}_{29}(J_{30}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30})\\},$
$\displaystyle\mathcal{B}_{30}(J_{31}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30})\\},$
$\displaystyle\mathcal{B}_{31}(J_{32}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30}),(u_{31},u_{32})\\},$
$\displaystyle\mathcal{B}_{32}(J_{33}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30}),(u_{31},u_{32}),(u_{32},u_{33})\\},$
$\displaystyle\mathcal{B}_{33}(J_{34}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30}),(u_{31},u_{32}),(u_{32},u_{33})\\},$
$\displaystyle\mathcal{B}_{34}(J_{35}(s_{3}))$ $\displaystyle=$
$\displaystyle\\{(u_{3},u_{4}),(u_{6},u_{7}),(u_{7},u_{8}),(u_{9},u_{10}),(u_{9},u_{11}),(u_{12},u_{13}),(u_{14},u_{15}),$
$\displaystyle(u_{14},u_{16}),(u_{17},u_{18}),(u_{19},u_{20}),(u_{19},u_{21}),(u_{22},u_{23}),(u_{24},u_{25}),$
$\displaystyle(u_{24},u_{26}),(u_{27},u_{28}),(u_{29},u_{30}),(u_{31},u_{32}),(u_{32},u_{33}),(u_{34},u_{35})\\}.$

Figure 4 depicts $J_{18}(s_{3})$.

Figure 4: $J_{18}(s_{3})$.

##  4 Application of a General Black Arc Algorithm

A general Black Arc Algorithm is not known yet. However, in the event of
finding such an algorithm, the complexity of application to energy graphs with
cut vertices can be simplified through partitioned application. It is known
that a vertex $u_{i}\in V(G)$ is a cut vertex of $G$ if and only if the set of
arcs $A(G)$ can be partitioned into subsets $A_{1}(G)$, $A_{2}(G)$ and the
induced arc-subgraphs $\langle A_{1}(G)\rangle$ and $\langle A_{2}(G)\rangle$
have only vertex $u_{i}$ in common.

######  Theorem 4.1.

Let the energy graph $G$ have $c$ cut vertices and the subsets $A_{1}(G),\\\
A_{2}(G),A_{3}(G),\ldots,A_{\ell}(G)$ of $A(G)$ form the maximal partition of
$A(G)$ such that all pairs of distinct induced arc-subgraphs $\langle
A_{i}(G)\rangle$ and $\langle A_{j}(G)\rangle$, where
$i,j\in\\{1,2,\ldots,\ell\\}$, have at most one vertex in common. Then,
$b^{\bullet}(G)=\sum\limits_{i=1}^{\ell}b^{\bullet}(\langle A_{i}(G)\rangle)$.

###### Proof.

Clearly, an induced arc-subgraph $\langle A_{i}(G)\rangle$ has no cut vertex
else, the arc partitioning is not a maximum. Furthermore, for those distinct
pairs of induced arc-subgraphs which have a vertex $u$ in common, such $u$ is
a cut vertex of $G$. It is observed that for any induced arc-subgraph $\langle
A_{i}(G)\rangle$ at least one induced arc-subgraph $\langle A_{j}(G)\rangle$
exists such that a common vertex $u\in V(G)$ between the arc-induced subgraphs
exists else, $G$ is disconnected. Consider any distinct pair of arc-induced
subgraphs $\langle A_{i}(G)\rangle$ and $\langle A_{j}(G)\rangle$ which share
the common vertex $u$.

Case 1: If $u$ is a source vertex in $G$ then $u$ is a source vertex in
$\langle A_{i}(G)\rangle$ and $\langle A_{j}(G)\rangle$, respectively. Hence
during the time interval $[0,1)$ of propagation no arc is rendered a black
arc.

Case 2: If $u$ is a sink vertex in $G$ then $u$ is a sink vertex in $\langle
A_{i}(G)\rangle$ and $\langle A_{j}(G)\rangle$, respectively. Let
$t^{\prime},t^{\prime\prime}$ be the total (exhaustive) propagation time
through $\langle A_{i}(G)\rangle$ and $\langle A_{j}(G)\rangle$, respectively.
Since a sink vertex can never initiate further propagation it cannot propagate
kinetic energy or particle mass which could dissipate. Hence, during the time
interval $[0,t^{\prime})$ of propagation the arcs in $\langle A_{i}(G)\rangle$
rendered black arcs are identical to those found in $G$. Similar reasoning
applies to $\langle A_{j}(G)\rangle$ over the propagation period
$[0,t^{\prime\prime})$.

Case 3: If $u$ is an intermediate ($d_{G}^{+}(u)>0$, $d^{-}_{G}(u)>0$) vertex
the independence of both propagation and the creation of black arcs lie in the
fact that for $u_{l}\in V(\langle A_{i}(G)\rangle)$ and $u_{k}\in V(\langle
A_{j}(G)\rangle)$ the arc $(u_{l},u_{k})\notin A(G)$. Therefore, the black
arcs resulting from propagation through $\langle A_{i}(G)\rangle$ during
$[0,t^{\prime})$ and those resulting from propagation through $\langle
A_{j}(G)\rangle$ during $[0,t^{\prime\prime})$ are independent in each induced
arc-subgraph and therefore identical to those found in $G$.

Therefore, $b^{\bullet}(G)=\sum\limits_{i=1}^{\ell}b^{\bullet}(\langle
A_{i}(G)\rangle)$, completing the proof. ∎

##  5 Conclusion

There is a wide scope for further research in respect of total dissipated
black energy and determining the black arc number for different classes of
graphs. The Jaco-type Black Arc Algorithm poses the challenge of complexity
analysis. Such analysis will contribute to theoretical computer science.
Determining the amount of kinetic energy which dissipate from an energy graph
$G$ is an open problem. Studying the properties of solid subgraphs is also
wide open. Some of the other open problems we have identified during our study
are the following.

######  Problem 1.

Verify whether the solid subgraph of an energy graph $G$ is connected.

######  Problem 2.

Find a closed formula for $b^{\bullet}(J_{n}(s_{3}))$ where $s_{3}=\\{$mod 5
sequence$\\}$, either in terms of the black arc number or in terms of
appropriate vertex indices.

######  Problem 3.

Describe a Black Arc Algorithm for a general energy graph.

######  Remark.

A Jaco-type graph can alternatively be defined as follows. A directed graph
$G$ is a Jaco-type graph if the vertex set is the set of positive integers and
there is a non-decreasing sequence $s=\\{a_{n}\\}_{n\in\mathbb{N}}$ of
positive integers such that a pair $(n,m)$ is an arc of $G$ if and only if
$n<m\leq a_{n}+n$.

An interesting, directed complement graph exists namely, the graph
$G^{\prime}$ with arcs $(n,m)$ such that $n<m$ and $(n,m)\notin A(G)$. Hence,
all pairs $(n,m)$ such that $a_{n}+n<m$ belong to $G^{\prime}$ so a strict
order on $\mathbb{N}$ is defined. Ordered sets obtained in this way have a
minimal type. It means the sets are infinite and no proper initial segment is
finite.

Conversely, a poset $P$ has minimal type if and only if elements can be
enumerated in a sequence$\\{v_{n}\\}_{n\in\mathbb{N}}$ and there is a sequence
$\\{a_{n}\\}_{n\in\mathbb{N}}$ of positive integers such that $v_{n}<v_{m}$ if
$n+a_{n}<m$. Also an interval order is an ordered st $P$ in which the poset
$Q$ made of the direct sum of two 2-element chains does not embed. If
moreover, $P$ does not embed the poset made of the direct sum of a 3-element
chain and a 1-element chain then $P$ is a semiorder. Also, for interval orders
both the predecessor and the successor order are total quasi-order. It implies
that a graph $G$ is a Jaco-type graph with enumeration $\\{v_{n}\\}_{n}$ if
and only if the order attached to the enumeration is the predecessor order
that corresponds to the order $G^{\prime}$ on the directed complement as
defined above. Finally, an order is minimal if and only if it extends the
directed complement of a Jaco-type graph.

A valid observation is that the sequence of positive integers need not
necessarily be non-decreasing. This observation and the perspective from a set
theory point of view opens further research.

## Acknowledgements

The authors of this article gratefully acknowledge the critical and
constructive comments of the anonymous referee, which significantly improved
the content and presentation of this article.

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