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# Anomaly Detection in Video Sequences: A Benchmark and Computational Model

Boyang Wan1 Wenhui Jiang1 Yuming Fang${}^{1\corr}$ Zhiyuan Luo1 Guanqun Ding1
<EMAIL_ADDRESS>1School of Information Management, Jiangxi University of
Finance and Economics, 665 Yuping West Street, Qingshanhu District, Nanchang
City, Nanchang, China

###### Abstract

Anomaly detection has attracted considerable search attention. However,
existing anomaly detection databases encounter two major problems. Firstly,
they are limited in scale. Secondly, training sets contain only video-level
labels indicating the existence of an abnormal event during the full video
while lacking annotations of precise time durations. To tackle these problems,
we contribute a new Large-scale Anomaly Detection (LAD) database as the
benchmark for anomaly detection in video sequences, which is featured in two
aspects. 1) It contains 2000 video sequences including normal and abnormal
video clips with 14 anomaly categories including  _crash, fire, violence_ ,
_etc._ with large scene varieties, making it the largest anomaly analysis
database to date. 2) It provides the annotation data, including video-level
labels (abnormal/normal video, anomaly type) and frame-level labels
(abnormal/normal video frame) to facilitate anomaly detection. Leveraging the
above benefits from the LAD database, we further formulate anomaly detection
as a fully-supervised learning problem and propose a multi-task deep neural
network to solve it. We firstly obtain the local spatiotemporal contextual
feature by using an Inflated 3D convolutional (I3D) network. Then we construct
a recurrent convolutional neural network fed the local spatiotemporal
contextual feature to extract the spatiotemporal contextual feature. With the
global spatiotemporal contextual feature, the anomaly type and score can be
computed simultaneously by a multi-task neural network. Experimental results
show that the proposed method outperforms the state-of-the-art anomaly
detection methods on our database and other public databases of anomaly
detection. Supplementary materials are available at
http://sim.jxufe.cn/JDMKL/ymfang/anomaly-detection.html.

##  1 Introduction

Anomaly detection, which attempts to automatically predict abnormal/normal
events in a given video sequence, has been actively studied in the field of
computer vision. As a high-level computer vision task, anomaly detection aims
to effectively distinguish abnormal and normal activities as well as anomaly
categories in video sequences. In the last few years, there have been many
studies investigating anomaly detection in the research community
Li2014Anomaly ; Cosar2017Toward ; Ravanbakhsh2017Abnormal ; Hasan2016Learning
; Liu2018GAN ; wan2020weakly ; park2020learning ; zaheer2020old ;
Liu2019MarginLE .

Compared with normal behaviors, an event that rarely occurs or with low
probability is generally considered as  _anomaly_. In practice, it is
difficult to build effective anomaly detection models due to the unknown event
type and indistinct definition of  _anomaly_. Traditionally, anomaly detection
methods are designed from two aspects. One type of anomaly detection method is
designed by reconstruction and they focus on modelling normal patterns in
video sequences Antic2011Video ; Ravanbakhsh2017Abnormal ; Hasan2016Learning ;
Lu2013Abnormal ; Liu2018GAN ; park2020learning ; zaheer2020old . The goal of
these methods is to learn a feature representation model for normal patterns.
At the testing stage, these methods utilize the differences between abnormal
and normal samples to determine the final anomaly score of testing data, such
as the reconstruction cost or specific threshold Ravanbakhsh2017Abnormal ;
Hasan2016Learning ; Lu2013Abnormal ; Liu2018GAN ; park2020learning ;
zaheer2020old . Although reconstruction-based anomaly detection methods are
good at reconstructing normal patterns in video sequences, the key issue in
these methods is that they rely heavily on training data.

() Fighting

() Boxing

() Hurt

() Touching

() Running-A

() Running-N

Figure 1: Definition of anomaly on different visual scenes, where Running-A and Running-N are short for Running Abnormal Events and Running Normal Events, respectively. Column (a) to (f): Fighting, Boxing, Hurt, Touching, Running Abnormal Events and Running Normal Events. Table 1: The detailed information of existing video anomaly detection databases. Database | Year | Videos | Scenes | Supervision | Categories
---|---|---|---|---|---
UCSD Ped1 Li2014Anomaly  | 2014 | 70 | 1 | Video-level | not specified
UCSD Ped2 Li2014Anomaly  | 2014 | 28 | 1 | Video-level | not specified
Avenue Lu2013Abnormal  | 2013 | 37 | 1 | Video-level | not specified
LV Leyva2017Video  | 2017 | 28 | 28 | Video-level | not specified
ShanghaiTech Luo2017A  | 2017 | 437 | 13 | Video-level | not specified
UCF-Crime Sultani2018Real  | 2018 | 1900 | - | Video-level | 13
LAD | - | 2000 | 1895 | Video- and frame- level | 14

Another type of anomaly detection method regards anomaly detection as a
classification problem Zhu2013Anomaly ; Colque2015Histograms . For these
methods, anomaly scores of video sequences are predicted by extracting
features such as Histogram of Optical Flow (HOF) or dynamic texture (DT) with
a trained classifier Zhu2013Anomaly ; Colque2015Histograms . The performance
of these methods is highly dependent on training samples. To obtain
satisfactory performance, extracting effective and discriminative features is
crucial for such anomaly detection methods.

Most of the existing anomaly detection methods are designed based on the
hypothesis that any pattern different from learned normal patterns is regarded
as an anomaly. Under this assumption, the same activity in different scenes
might be denoted as a normal or abnormal event. For example, as shown in Fig.
1, a fighting scene where two men are brawling may be considered as abnormal,
while it may be normal when these two men are doing boxing sport; a girl/boy
running on the street because of panic may be considered as abnormal, but it
may be normal when the weather is raining since the girl/boy forget to take an
umbrella; an animal touching human may be considered as abnormal (i.e., snake
bite human), while it may be normal in the case of kissing people by a
dolphin. Additionally, there is much redundant visual information in high-
dimensional video data, which increases the difficulty for event
representation in video sequences.

The main challenges of anomaly detection task are caused by the lack of a
large-scale anomaly detection database with fine-grained annotations. Although
several anomaly detection databases are proposed in the research community
Li2014Anomaly ; Lu2013Abnormal ; Leyva2017Video ; Sultani2018Real ; Luo2017A ,
they are flawed either in the dataset scale or annotation richness.
Specifically, there are no more than 100 video sequences in Li2014Anomaly ;
Lu2013Abnormal ; Leyva2017Video , which could not satisfy the requirement of
training data for deep learning based models. Besides, existing databases
Li2014Anomaly ; Lu2013Abnormal ; Leyva2017Video ; Sultani2018Real ; Luo2017A
only provide video-level labels in training set, which makes it infeasible to
learn anomaly detection models in a fully-supervised manner. Moreover, the
definition of anomaly is unclear, which makes it hard for anomaly ground-truth
annotation and computational model design. For anomaly detection models for
specific events such as hyperspectral anomaly detection yuan2015hyperspectral
, violence detector Mohammadi2016Angry  and traffic detector
Sultani2010Abnormal ; yuan2016anomaly , their applications are limited since
they cannot be used to detect other abnormal events.

To address these above problems in existing anomaly detection studies, we
investigate anomaly detection from the following two aspects in this study.

  * •

We build a new Large-scale Anomaly Detection (LAD) database consisting of
$2000$ video sequences and corresponding anomaly ground-truth data including
video-level labels (abnormal/normal video, anomaly type) and frame-level
labels (abnormal/normal video frame). There are 14 abnormal event categories
in total. More than 100 video sequences are collected for each abnormal
category, making it the largest database for anomaly detection to date.

  * •

We propose a multi-task deep neural network for anomaly detection by learning
local and global contextual spatiotemporal features with a multi-task joint
learning scheme. An inflated 3D convolutional network is constructed to
extract local spatiotemporal contextual features, which are further used to
input a designed recurrent convolutional neural network to learn global
spatiotemporal contextual features. The anomaly category and score can be
predicted by a multi-task deep network with these global features.

The rest of this paper is organized as follows. Section II reviews the related
work. Section III provides the details of the built large anomaly database,
including the data collection and annotations. Section IV describes the
proposed method in detail. Section V briefly describes the performance
evaluation metrics and the performance comparison of the proposed method. We
conclude this paper in Section VI.

##  2 Related Work

###  2.1 Anomaly Detection Databases

Currently, there have been several anomaly detection databases for video
sequences Li2014Anomaly ; Lu2013Abnormal ; Leyva2017Video ; Leyva2017Video ;
Sultani2018Real . The detailed information of these existing databases is
given in Table 1.

UCSD Li2014Anomaly  includes two subsets of Ped1 and Ped2, where an anomaly
event is defined as a car or a bicycle appearing abnormally in the street
compared with normal patterns of the car or pedestrian. In this database, the
crowd density of different video sequences is different. All video sequences
are with 10 Frames Per Second (fps), including two different outdoor scenes.
The first subset Ped1 contains 34 training and 36 testing video sequences,
including around 8000 video frames in total, while the second subset Ped2
contains 16 training and 12 testing video sequences including 4950 video
frames in total.

Avenue Lu2013Abnormal  contains 16 training and 21 testing video sequences. In
this database, abnormal events are labeled as people running, loitering,
throwing,  _etc._ The size of a person may vary depending on the position and
angle of the camera. It provides pixel-level annotation for each video frame.
Each video sequence is about 2 minutes long. There are around 31000 video
frames with a resolution of $640\times 360$ in total. All video sequences are
recorded in the same visual scene.

LV Leyva2017Video  This database contains 28 realistic video sequences for
out-door and in-door scenes, and abnormal events are labeled as people
fighting, people clashing, vandalism,  _etc._ Each video sequence is divided
into training and testing data.

ShanghaiTech Luo2017A  contains 437 realistic video sequences for out-door
scenes. There are 13 different visual scenes in this database, where all video
sequences are captured by surveillance cameras. This database 130 abnormal
events including Running, bicycles, skaters  _etc._

UCF-Crime Sultani2018Real  contains 13 real-world anomaly categories,
including  _Abuse, Arrest, Arson, Assault, Accident, Burglary, Explosion,
Fighting, Robbery, Shooting, Stealing, Shoplifting_ and  _Vandalism_. It
includes 1900 surveillance video sequences in total, composed of 950 abnormal
video sequences and 950 normal video sequences. There are about 128 hours for
all these video sequences in this database. The testing set including 150
normal and 140 abnormal video sequences, while the rest are used as the
training set. This database provides only video-level labels for training
videos.

From Table 1, we can observe that the training video of most existing anomaly
detection databases is limited in scale. Although they contain a variety of
abnormal events, the categories of abnormal videos are not specified. However,
the visual scenes in the real world are diverse and complicated with different
anomaly types. Another common drawback of existing databases is the lacking of
frame-level labels. As a result, anomaly detection algorithms can only be
learned in a weakly-supervised manner, which deteriorates the performance and
impedes the wide usage in practical applications.

In this work, we build a new large-scale anomaly detection database, including
2000 video sequences and the corresponding video- and frame-level labels, to
promote anomaly detection in a fully-supervised manner. The built database
contains 1895 different visual scenes with 14 anomaly categories, including
_Crash, Crowd, Destroy, Drop, Falling, Fighting, Fire, Fall Into Water, Hurt,
Loitering, Panic, Thiefing, Trampled,_ and  _Violence_. We will introduce this
database in detail in Section III.

###  2.2 Anomaly Detection Methods

Early anomaly detection studies extract object trajectories to detect abnormal
activities in video sequences, where an object against the learned normal
object trajectories is detected as an anomaly Cosar2017Toward ;
Piciarelli2008Trajectory ; Wu2010Chaotic ; piciarelli2006on-line ;
jiang2011anomalous ; tung2011goal-based ; morris2011trajectory ;
calderara2011detecting ; patino2015abnormal ; yi2015understanding . Cosar  _et
al._ proposed an unsupervised architecture for abnormal behavior prediction by
object trajectory analysis (i.e., speed, direction, and body movement) and
pixel-level analysis (i.e., appearance) Cosar2017Toward . Piciarelli  _et al._
designed an anomaly detection model by clustering extracted normal
trajectories of moving objects in video sequences Piciarelli2008Trajectory ;
piciarelli2006on-line . Specifically, they utilized a single-class SVM to
learn normal object trajectories. In the testing stage, a new trajectory is
predicted as an anomaly or not by comparing it with the clustering model with
a threshold. Wu  _et al._ exploited chaotic invariants of lagrangian particle
trajectories to represent anomaly activities in crowded scenes Wu2010Chaotic .
Patino  _et al._ detected speed and direction change by trajectories of moving
objects, which predict anomaly events patino2015abnormal . Jiang  _et al._
proposed a context-aware anomaly detection method jiang2011anomalous . By
tracking all moving objects in a video sequence, the anomaly event is detected
by considering different levels of spatiotemporal contexts. Morris  _et al._
studied the features of the normal recurrent motion patterns of the
surveillance subjects to detect abnormalities morris2011trajectory . Yi  _et
al._ proposed a pedestrian behavior model for anomaly detection by stationary
crowd group yi2015understanding . However, these methods can not work well
when objects are occlusive.

To solve the challenging problem from object occlusion, some studies used
global features to represent complex scenes for anomaly detection
Li2014Anomaly ; Antic2011Video ; Leyva2017Video ; Wang2014Detection ;
Mehran2009Abnormal ; adam2008robust ; saligrama2012video ;
benezeth2009abnormal ; kim2009observe ; kratz2009anomaly ; zhang2005semi-
supervised ; roshtkhari2013online ; zhu2013context-aware ; xiao2015learning ;
Cui2011Abnormal ; Yuan2015Online ; Cheng2015Gaussian . Then they used a
nonlinear one-class support vector machine to learn normal patterns. The event
behavior with an outlier score predicted by the trained model is considered an
an anomaly. Different from the study Wang2014Detection , Li  _et al._ proposed
a joint anomaly detection model by combining temporal and spatial anomalies
with a Mixture of Dynamic Textures (MDT) for modelling normal crowd activities
Li2014Anomaly . Besides, Mehran  _et al._ introduced a social force model to
stimulate the normal behaviour of the crowd. Then they classified video frames
as normal or abnormal by using a bag of words approach Mehran2009Abnormal .
Cui  _et al._ defined a concept of interaction energy to represent the current
interaction between the surrounding region and objects. A behaviour is
considered as anomaly when the energy and velocity of an object change
dramatically Cui2011Abnormal . Adam  _et al._ used low-level information based
on multiple local monitors for anomaly detection in video sequences
adam2008robust . In order to detect abnormal events in video sequences,
Saligrama  _et al._ used spatiotemporal features with a $k$-nearest neighbor
method to design an anomaly detection model saligrama2012video . Benezeth  _et
al._ used normal events to train a spatiotemporal co-occurrence matrix and
used the matrix and Markov random field to detect anomaly benezeth2009abnormal
. Kim  _et al._ used a mixture of probabilistic PCA models to present the
local optical flow pattern, and used the representation and Markov random
field to define normal patterns kim2009observe . Antic  _et al._ introduced a
probabilistic model by localizing abnormalities with statistical inference
Antic2011Video . Yuan  _et al._ proposed an informative Structural Context
Descriptor (SCD) to describe the crowd scene for anomaly detection
Yuan2015Online . Lu  _et al._ proposed to learn multiple dictionaries to model
normal patterns with sparse constraint Lu2013Abnormal . Leyva  _et al._
designed an anomaly detection method based on optical flow information and
foreground occupancy Leyva2017Video . In athanesious2020detecting , a novel
hand-craft optical-optical feature extractor named Super Orientation Optical
Flow (SOOF) is proposed to efficiently capture motion information of objects
in surveillance videos. In yuan2015hyperspectral , a vertex- and edge-weighted
graph is constructed to reduce false-positive rate in hyperspectral anomaly
detection task. To tackle specific problems caused by dynamic outdoor
environments in traffic scenes, Yuan  _et al._ proposed a spatial localization
constrained sparse coding approach as a motion descriptor.

Recently, deep learning techniques have been widely used to build anomaly
detection models Ravanbakhsh2017Abnormal ; Hasan2016Learning ; Liu2018GAN ;
Sultani2018Real ; Sabokrou2017Deep ; sabokrou2016video ; Luo2017Remembering ;
Hinami2017Joint ; Luo2017A ; Ionescu2017Unmasking ; Xu2015learning . Sabokrou
_et al._ proposed a cascaded Deep Neural Networks (DNN) for anomaly detection
by hierarchically modelling normal patches using deep features, then they used
Gaussian classifier to identify abnormal behaviours in video sequences
Sabokrou2017Deep . Ravanbakhsh  _et al._ trained two Generative Adversarial
Nets (GANs) to learn normal patterns in video sequences
Ravanbakhsh2017Abnormal . During the training stage, the first generator of
GANs takes a normal video frame as input and produces a reconstructed optical
flow image, while the second generator of GANs is fed into a real optical-flow
image and generates a reconstructed appearance image. In the testing stage,
this model detects anomalies by using the reconstruction differences between
real data (original video frames and original optical-flow images) and
generated data (reconstructed video frames and reconstructed optical-flow
images. Hasan  _et al._ proposed two auto-encoder models to learn temporal
regularity for anomaly detection Hasan2016Learning . Similarly, Xu  _et al._
proposed a deep neural network based model by constructing a stacked denoising
autoencoder for feature learning for abnormal event detection Xu2015learning .
Luo  _et al._ proposed a Convolutional LSTMs Auto-Encoder (ConvLSTM-AE) to
encode normal appearance and motion patterns for abnormal event detection
Luo2017Remembering . Hinami  _et al._ learned a Convolutional neural network
through multiple visual tasks, then they used semantic information to detect
anomaly events Hinami2017Joint . Ionescu  _et al._ applied the unmasking
technique to train a binary classifier to distinguish two consecutive short
video sequences and gradually remove the most discriminant features
Ionescu2017Unmasking . Luo  _et al._ proposed a Temporally-coherent Sparse
Coding (TSC) approach for anomaly detection, in which similar adjacent frames
are encoded with similar reconstruction coefficients Luo2017A . Liu  _et al._
proposed an anomaly detection model based on the difference between a
predicted frame and the ground-truth, where the temporal constraint is
considered besides spatial constraints Liu2018GAN . Sultani  _et al._ learned
a generic model using deep Multiple Instance Learning (MIL) framework with
weakly labeled data Sultani2018Real , and Wan  _et al._ proposed a dynamic MIL
loss and a center loss for enlarging the inter-class distance between
anomalous and normal instances and reducing the intra-class distance of normal
instances, respectively.

All above deep learning based methods formulate anomaly detection as the
unsupervised learning or weakly-supervised learning problem due to the lack of
frame-level labels in the training set of the existing anomaly detection
databases. In this paper, leveraging the fine-grained frame-level annotation
from our proposed LAD database, we formulate anomaly detection as a fully-
supervised learning problem and propose a novel multi-task deep neural network
to address anomaly detection in videos. Through extensive experimental
analysis, we show that our model significantly improves the performance
anomaly detection.

##  3 Anomaly Detection Benchmark

###  3.1 Data Collection

Crash Crowd

Destroy Drop

Falling FallIntoWater

Fighting Fire

Hurt Loitering

Panic Thiefing

Trampled Violence

Normal (I) Normal (II)

Figure 2: Visual samples of different anomaly event categories in the proposed
LAD database. The proposed database contains 14 distinct anomaly categories,
including  _Crash, Crowd, Destroy, Drop, Falling, Fall Into Water, Fighting,
Fire, Hurt, Loitering, Panic, Thiefing, Trampled and Violence,_ as well as
_Normal_ activities.

To collect large-scale representative anomaly activities, we search for a
large number of video sequences from public websites including
YouTube111https://www.youtube.com/, YouKu222https://www.youku.com/, and
Tencent Video333https://v.qq.com/. Besides, we collect some video sequences
from existing activity recognition databases, such as FCVID FCVID , Hollywood2
Marszalek2009Actions , and YouTube Action Liu2009Recognizing . Additionally,
we record some normal activities or suddenly occurring abnormal events in the
square and school by a digital camera to provide plenty of visual scenes and
real-world events. With these operations, we initially collect over 2500 video
sequences in total.

We analyze the collected video sequences and classify these video sequences
into 14 categories, including  _Crash, Crowd, Destroy, Drop, Falling,
Fighting, Fire, Fall Into Water, Hurt, Loitering, Panic, Thiefing, Trampled,_
and  _Violence_. For each category, we discard some video sequences which fall
into any of the following two conditions: (1) low resolution or low quality;
and (2) incomplete anomaly event or anomaly is not clear. We strictly select
more than 100 video sequences, including more than 50 normal video sequences
and 50 abnormal video sequences for each category. Finally, we preserve 14
distinct anomaly categories with 2000 video sequences totally. The frame rate
of all video sequences is 25 fps. For each video sequence, we manually extract
a video segment that represents an abnormal/normal activity by irrelevant
video frames. In Fig. 2, we show four frames of an example video for each
anomaly category, including 2 normal frames and 2 abnormal frames.

###  3.2 Annotations

Figure 3: The statistics information of the proposed LAD database. (a) The
anomaly distribution of each anomaly category; (b) the anomaly distribution of
each video sequence; (c) the number of video frames for each video sequence.

As a high-level video analysis task, anomaly detection requires frame-level
labels to identify the time period of an abnormal event starts and video-level
labels to recognize the anomaly category. Thus, we provide both video- and
frame-level labels in our database. To ensure the quality of annotations, we
invite five postgraduate students to take part in our annotation experiment.
In the annotation experiment, we define 1 as abnormal video frame and 0 as
normal video frame. We first ask annotators to find the video frames where an
anomaly event begins and ends, which are all labeled as 1, and the rest are
labeled as 0. Then, we compute the averaging scores of annotations for each
frame. Finally, we binaries the averaging scores by using threshold 0.5, and
take binary averaging scores as the frame-level anomaly labels. Video-level
labels are annotated to represent anomaly category, where a video sequence is
labeled anomaly if any frame in this video sequence is abnormal.

In this database, a normal video sequence in each anomaly category denotes
that behaviour in this video sequence is regarded as normal. For example, for
the  _Fighting_ category, the boxing activity is classified as normal though
it is similar to the fighting anomaly event; for the  _Falling_ category, a
woman falling down when playing roller-skating is labeled as anomaly, while a
woman bending into a squat with knees is annotated as normal; for the  _Hurt_
category, a woman being attacked by a dog is labeled as anomaly, while a woman
walking the dog is annotated as normal. We divide the built database into
training and testing subsets. The testing set contains 560 sequences, composed
of randomly selected 20 abnormal and 20 normal video sequences for each
anomaly category. The rest are used as training set. The statistics
information of all video sequences is shown in Fig. 3. In the built database,
we record the entire process from the beginning to the end for anomaly events,
and we use a video sequence to represent a complete event. As shown in Fig. 3,
the number of video frames for most video sequences is in the range of
$\left[4000,8000\right]$. The anomaly percentage of Fire and Loitering event
categories are high because the anomaly of these anomaly events generally
lasts for a long time. The video frames with smoke or small fires are
considered anomalies when we annotate abnormal fire frames for Fire. By
contrast, the anomaly percentage of the Falling category is the lowest since
this type of anomaly event lasts for a short time. In this type of video
sequences, when a person falls down, he can stand up quickly. Besides, we
compare our database with UCF-Crime Sultani2018Real , and find that there are
some video sequences with an anomaly percentage higher than 0.5. Since the
anomaly events of these video sequences last for a long time, the whole event
can be fully expressed. In addition, the abnormal frames of UCF-Crime database
Sultani2018Real  are not completely labeled. For example, the authors only
consider the moment of explosion as abnormal for the Explosion category, but
the fire generated after the explosion is regarded as normal.

##  4 Proposed Method

Figure 4: The architecture of the proposed anomaly detection method by
modeling local and global spatiotemporal contextual features.

Here, we propose a multi-task deep neural network for anomaly detection. The
proposed model is demonstrated in Fig. 4. It consists of two components, i.e.,
a local and a global spatiotemporal context-aware streams.

Our observation is that local outliers may failed to extract feature
representation of continuous action. To alleviate this problem, we devise a
local spatiotemporal context aware submodule and a spatiotemporal context
aware submodule, as shown in Fig. 4. In particular, we first encode each video
sequence by feature representation with a pretrained Inflated 3D convolutional
network (I3D) CarreiraZ17 . Given a video sequence with $M$ frames, we divide
it into $N$ clips, and each clip contains $m$ video frames. Thus, each video
sequence can be denoted as
${\bm{V}}=\left\\{{{\bm{v}}_{n},N=M/m}\right\\}_{n=1}^{N}$. The split clips
are fed into a pretrained I3D to extract high-level visual features. For $K$
consecutive clips, the local feature vectors can be represented as
$\bm{X}=\left\\{{\bm{x}}_{t}\right\\}_{t=1}^{K}$.

The video sequence is high dimensional data that contains plenty of visual
information. Thus, preserving important cues yet filtering out the redundancy
is important to learn effective anomaly detection models. To learn robust
global spatiotemporal contextual cues, we feed the extracted local contextual
features of $K$ consecutive clips into the global context-aware stream to
learn high-level features. As shown in Fig. 4, we adopt a two-layer
Convolutional LSTM (ConvLSTM) network Shi2015Convolutional  to learn global
spatiotemporal features of a video segment. Unlike LSTM, ConvLSTM
Shi2015Convolutional  is designed by using three-dimensional data as the input
and uses convolutional operation, which can obtain temporal information and
extract spatial features. At the same time, it provides good generalization by
reducing the number of parameters and the computational complexity.
Specifically, we show the formula for ConvLSTM as follows.

${\bm{{i}}}_{t}=\sigma(\bm{W}_{xi}*\bm{X}_{t}+\bm{W}_{hi}*\bm{H}_{t-1}+\bm{W}_{ci}\circ\bm{C}_{t-1}+b_{i})$
(1)
${\bm{{f}}}_{t}=\sigma(\bm{W}_{xf}*\bm{X}_{t}+\bm{W}_{hf}*\bm{H}_{t-1}+\bm{X}_{cf}\circ\bm{C}_{t-1}+b_{f})$
(2)
$\bm{C}_{t}={\mathclap{\bm{{f}}}_{t}}{\circ}{\bm{C}_{t-1}}+{\mathclap{\bm{{i}}}_{t}}\circ{\rm
tanh}(\bm{W}_{xc}*\bm{X}_{t}+\bm{X}_{hc}*\bm{H}_{t-1}+b_{c})$ (3)
${\bm{{o}}}_{t}=\sigma(\bm{W}_{xo}*\bm{X}_{t}+\bm{W}_{ho}*\bm{H}_{t-1}+\bm{W}_{co}\circ\bm{C}_{t}+b_{o})$
(4) $\bm{H}_{t}={\bm{{o}}}_{t}\circ{\rm tanh}(\bm{C}_{t})$ (5)

where ${\bm{{X}}}_{t}$ and $\bm{H}_{t}$ denote the input and output of
ConvLSTM Shi2015Convolutional  at time $t$; ${\bm{{i}}}_{t}$,
${\bm{{f}}}_{t}$, ${\bm{{o}}}_{t}$ and $\bm{C}_{t}$ represent outputs of input
gate, forget gate, output gate and memory cell; $*$ is a convolutional
operation; $\circ$ represents the Hadamard product; and $\sigma$ is the
sigmoid activation function.

For ConvLSTM, we use the local features extracted from each clip of a video
segment as the input. The ConvLSTM network leverages both long- and short-term
cues of input features. The hidden states of the last layer of ConvLSTM are
fed into three fully convolutional layers to predict the final event category
and anomaly scores.

Furthermore, we design a multi-task joint learning network for learning the
intrinsic relationship between anomaly detection and classification. The sub-
network of anomalous categories classification task is designed to recognize
anomaly category, and we use a cross-entropy loss function in this sub-
network.

$\emph{$L$}_{1}=-{\sum_{i=1}^{C}}{\hat{y}_{i}}\log{{y}_{i}}+\gamma\parallel\bm{W}\parallel_{2}^{2}$
(6)

where ${{\hat{\bm{y}}}}=[\hat{y}_{1},\hat{y}_{2},...,\hat{y}_{C}]$ denotes the
one hot encoding of anomalous category label for a video sequence;
${\bm{y}}=[{y}_{1},{y}_{2},...,{y}_{C}]$ represents the corresponding score
vector predicted by the sub-network of anomalous categories classification;
$\parallel{\bm{W}}\parallel_{2}^{2}$ is a $L_{2}$-norm regularization term to
avoid over-fitting; $\gamma$ is a hyper-parameter to balance the trade-off
between the loss and regularization.

As the sub-task of anomaly score prediction is modeled as a regression
problem. We use $smooth$ loss function Girshick2015Fast  as learning objective
in this sub-network as follows.

$\begin{split}&~{}\emph{$L$}_{2}=\sum_{i}({smooth}(s_{i}-\hat{s_{i}}))\\\
&\emph{smooth}(x)=\begin{cases}0.5x^{2},&\left|x\right|\leq
1\cr\left|x\right|-0.5,&otherwise\end{cases}\end{split}$ (7)

where $\hat{s_{i}}$ denotes the anomalous label of a video frame; ${s_{i}}$
represents the corresponding score predicted by the sub-network of anomaly
score prediction. Based on ${L}_{1}$ and ${L}_{2}$, the final loss function is
written as follows:

$\emph{$L$}={\lambda}_{1}{L}_{1}+{\lambda}_{2}{L}_{2}$ (8)

where $\lambda_{1}$ and $\lambda_{2}$ are hyper-parameters to weight the
importance of two sub-task.

##  5 Experimental Results

Table 2: Data splits on Avenue Lu2013Abnormal , UCSD Ped2 Li2014Anomaly , UCF-Crime Sultani2018Real , ShanghaiTech Luo2017A  and LAD databases. Split | Subset | Avenue Lu2013Abnormal  | UCSD Ped2 Li2014Anomaly  | ShanghaiTech Luo2017A  | UCF-Crime Sultani2018Real  | LAD
---|---|---|---|---|---|---
Unsupervised | Train | 8 | 8 | 175 | 800 | 958
Test | 18 | 14 | 199 | 290 | 560
Weakly-supervised | Train | 19 | 14 | 238 | 1610 | 1440
Test | 18 | 14 | 199 | 290 | 560
Fully-supervised | Train | 19 | 14 | 238 | 1610 | 1440
Test | 18 | 14 | 199 | 290 | 560

###  5.1 Implementation and Evaluation Metrics

Implementation: In this work, the proposed deep anomaly detection network is
implemented in Ubuntu operating system with Tensorflow 65Abadi2016 . The
experiments are conducted with Intel Core I7-6900K*16 CPU (3.20GHz), 64 GB
RAM, and Nvidia TITAN X (Pascal) GPU with 16 GB memory.

The I3D is pretrained by using Kinetics-400 CarreiraZ17 , which is a large-
scale video classification dataset. We set the ${\lambda}_{1}$ and
${\lambda}_{2}$ as 1 and 10, respectively. We use a threshold of 0.5 to obtain
the binarization anomaly score. Here, we use Adam optimizer Kingma2014  to
update parameters in the proposed model, the learning rate is set as 3e-4, the
weight decay is set as 5e-4, the batch size is set as 60.

We divide the video sequence into clips with $m$=16 consecutive non-overlap
video frames and set $K$=5. A total of $K$ $\times$ $m$=80 frames are used as
the input to I3D to extract local spatiotemporal contextual features. The
output of the sub-network of event classification is set as $C$=14, which is
the number of LAD anomaly categories. We extract a $p$=1024 dimension feature
of last pooling layer in the I3D, and concatenate the outputs of RGB and
optical-flow I3D as the local spatiotemporal contextual feature of a video
clip. The video frames are resized to $224\times 224$, with their mean values
being removed.

The channel of hidden layers is set as 128 in proposed two-layer ConvLSTM,
with $3\times 3$ convolutional kernel and $1\times 1$ stride. The dimension of
the output features in our ConvLSTM is $4\times 4\times 128$. After reshaping
the learned global spatiotemporal contextual feature as a 2048-dimension
vector, we feed it into four fully convolutional layers for the final anomaly
score prediction. The dimensions of the first three fully convolutional layers
are set as 2048, 1024, and 512, respectively. The last layers are set as
$C$=14 and $m\times K$=80 for anomalous categories classification and score
prediction, respectively.

Table 3: AUC results on Avenue Lu2013Abnormal , UCSD Ped2 Li2014Anomaly , UCF-Crime Sultani2018Real , ShanghaiTech Luo2017A  and LAD databases, where $\mathcal{U}$, $\mathcal{W}$ and $\mathcal{S}$ represent unsupervised, weakly-supervised and fully-supervised methods, respectively. The * indicates experimental results are performed by using public source code. |  | Avenue Lu2013Abnormal  | UCSD Ped2 Li2014Anomaly  | ShanghaiTech Luo2017A  | UCF-Crime Sultani2018Real  | LAD
---|---|---|---|---|---|---
Sparse Lu2013Abnormal  | $\mathcal{U}$ | - | - | - | 65.51 | 50.31*
ConvAE Hasan2016Learning  | $\mathcal{U}$ | - | - | - | 50.60 | 53.24*
GMM Leyva2017Video  | $\mathcal{U}$ | - | - | - | 56.43* | 41.02*
Stacked RNN Luo2017A  | $\mathcal{U}$ | 70.09* | 52.58* | 67.66* | - | 49.42*
U-Net Liu2018GAN  | $\mathcal{U}$ | 55.26* | 71.26* | 56.59* | - | 53.96*
MNAD park2020learning  | $\mathcal{U}$ | 73.58* | 46.72* | 51.13* | 56.20* | 45.84*
OGNet zaheer2020old  | $\mathcal{U}$ | 63.23* | 69.08* | 69.26* | - | 55.07*
DeepMIL Sultani2018Real  | $\mathcal{W}$ | 87.53* | 90.19* | 86.30 | 75.41 | 70.18*
MLEP Liu2019MarginLE  | $\mathcal{W}$ | 89.20 | - | 73.40 | 50.01* | 50.57*
AR-Net wan2020weakly  | $\mathcal{W}$ | 89.31* | 93.64* | 91.24 | 74.36* | 79.84*
Our Method | $\mathcal{F}$ | 89.33 | 95.12 | 92.97 | 74.98 | 86.28

Evaluation Metrics: In this study, following existing anomaly detection
studies Luo2017A ; Liu2019MarginLE , we utilize a frame-level Area Under ROC
(Receiver Operating Characteristic) Curve (AUC) for quantitative performance
evaluation. A higher AUC value indicates better performance. To evaluate
anomalous categories performance of our model, we use the accuracy as the
metric.

Data Splits:We conduct the experiments upon six databases including Avenue
Lu2013Abnormal , UCSD Ped2 Li2014Anomaly , ShanghaiTech Luo2017A , UCF-Crime
Sultani2018Real  and LAD. We adopt three data splits of each database to fit
requirements of unsupervised, weakly-supervised and fully-supervised anomaly
detection methods.

Weakly-supervised Splits In the standard protocol of Avenue, UCSD Ped2,
ShanghaiTech, all training videos are normal, and this sitting is not suit for
weakly-supervised learning. So we reorganize these databases. For Avenue, UCSD
Ped2, we selected randomly 50% video to be training video, while the rest are
used as the testing set. We use the same splits from wan2020weakly  and
Sultani2018Real  for ShanghaiTech and UCF-Crime, respectively. Only video-
level anomalous label is provided for training weakly-supervised anomaly
detection models.

Unsupervised Splits For each database, we use only normal videos in training
set of the weakly-supervised split to train unsupervised anomaly detection
models, and evaluate the unsupervised anomaly detection models by using all
videos in test set of the weakly-supervised split.

Fully-supervised Splits We use the same data splits of weakly-supervised
methods. Frame-level anomalous label and video-level anomalous categories
label are provided for training model. It worthy notice that UCF-Crime does
not provide frame-level anomalous label for the default training set, we use
video-level anomalous label to be frame-level anomalous label for each
training video.

The numbers of training and testing videos on different splits of the
databases are shown in Table 2.

###  5.2 Comparison with State-of-the-art Models

In this section, we compare the proposed approach with several state-of-the-
art unsupervised and weakly-superivsed anomaly detection methods. The
unsupervised anomaly detection methods contain GMM Leyva2017Video , Sparse
Lu2013Abnormal , ConvAE Hasan2016Learning , Stack RNN Luo2017A , U-Net
Liu2018GAN , MNAD park2020learning  and OGNet zaheer2020old . The weakly-
superivsed anomaly detection methods contain DeepMIL Sultani2018Real , MLEP
Liu2019MarginLE  and AR-Net wan2020weakly .

GMM is an anomaly detection model by using Gaussian Mixture Model and Markov
Chains. Sparse is a dictionary-based anomaly detection model by learning
normal dictionary using sparse representation. ConvAE is the first deep
learning based anomaly detection model by using Auto-Encoder to model normal
event patterns. Stack RNN U-Net is an anomaly detection model based on the
reconstruction errors between a predicted frame and the ground-truth. MNAD and
OGNet are latest unsupervised anomaly detection methods. We retrain these
models by using unsupervised splits on each database in this paper. DeepMIL is
a representative weakly-supervised anomaly detection method, and AR-Net
achieves the highest AUC performance so far in ShanghaiTech. The performance
of weakly-supervised anomaly detection methods are obtained by using weakly-
supervised splits on each database. It should notices that unsupervised
anomaly detection methods only use the normal videos to train their models.

We show the comparison results in Table 3 on Avenue, UCSD Ped2, ShanghaiTech,
UCF-Crime and LAD. Our method outperforms all competing anomaly detection
models on our LAD and achieves an absolute gain of 6.44% in terms of AUC,
compared to the state-of-the-art wan2020weakly . Compare the weakly-supervised
methods, our method achieves similar AUC performance on UCF-Crime. It may be
caused by using noisy frame-level anomalous label, which is the same as video-
level anomalous label, to train our model. Our model achieves higher AUC
performance than the competing anomaly detection models on Avenue, UCSD Ped2
and ShanghaiTech, it reveals frame-level annotation is an effective tool to
promote anomaly detection task.

As shown in Table 3, the competing weakly-supervised anomaly detection models
obtain higher AUC performance on Avenue, UCSD Ped and ShanghaiTech, compare to
unsupervised anomaly detection models. It indicates that the database
including training abnormal videos is necessity to promote the video anomaly
detection task. Most competing models achieve higher AUC performance on Avenue
and UCSD Ped, while they get relative lower AUC performance on ShanghaiTech,
UCF-Crime and LAD. These experimental results demonstrate that variety of
visual scenes is a indeterminable issue to current anomaly detection models.
It indicates that our LAD, which contains thousands visual scenes, is a
challenge database for video anomaly detection.

Table 4: Competition of different local spatiotemporal feature extractor,
where R
and O indicate RGB and Optical-flow, respectively. | Input modal | AUC
---|---|---
C3D Du2015Learning  | R | 77.21
$\rm I3D^{RGB}$ CarreiraZ17  | R | 84.43
$\rm I3D^{Optical-flow}$ CarreiraZ17  | O | 82.46
I3D CarreiraZ17  |  R&O | 86.93

###  5.3 Ablation Study

To evaluate the effectiveness of the local spatiotemporal feature extractor,
we compare four different spatiotemporal networks including C3D Du2015Learning
, $\rm I3D^{RGB}$ CarreiraZ17 , $\rm I3D^{Optical-flow}$ CarreiraZ17  and I3D
CarreiraZ17  on LAD. As shown in Table 4, our method with C3D achieves a
frame-level AUC of 77.21%. And the $\rm I3D^{RGB}$ and $\rm I3D^{RGB}$ based
our method achieves 84.43% and 82.46% in terms of AUC, respectively. Our
Method with I3D boosts the performance with a frame-level AUC of 86.93%.

The comparison results by using different loss functions in Table 5 illustrate
the boost brought by the proposed multi-task loss functions. Our method with
$L_{1}$, a loss function for the anomaly score prediction task, is treated as
the baseline. It achieves a frame-level AUC of 80.43% on LAD. While our method
with both $L_{1}$ and $L_{2}$ obtains an absolute gain of 6.50% in terms of
AUC. And the $L_{1}$ loss function boosts the anomalous categories performance
by obtaining a 59.3% in terms of accuracy.

Table 5: AUC and accuracy results obtained by using different loss functions
on LAD. $L_{1}$ | $L_{2}$ | AUC | Accuracy
---|---|---|---
✓ | ✕ | 80.43 | 3.1
✕ | ✓ | 50.00 | 58.4
✓ | ✓ | 86.93 | 59.3

###  5.4 Qualitative Analysis

Figure 5: AUC of different $K$ values.

In order to gain insight into the hyperparameter $K$, we perform experiments
using the I3D local spatiotemporal feature extractor with different values of
$K$, as shown in Fig. 5. Our method achieves the best performance in terms of
AUC when we set $K=5$, and our method with $K=2$ obtains an absolute reduction
of 3.97% in terms of AUC. It confirms the necessity to gain global
spatiotemporal features. The comparison results by using different
$\lambda_{2}$ values are shown in Fig. 6, and we set $\lambda_{1}$=1 in these
experiments. Our method gains the boost when $\lambda_{2}$ is set as 10.

Figure 6: AUC of different $\lambda_{2}$ values. Figure 7: Visualization of the confusion matrix of anomalous categories classification results by using our method.  Table 6: Experimental results of anomalous categories classification. | UCF-Crime | LAD
---|---|---
TCNN hou2017tube  | 28.4 | -
C3D Du2015Learning  | 23.0 | 45.9
Our Method | 59.6 | 59.3

As shown in Table 6, we can observe that our method outperforms the competing
model on UCF-Crime and LAD. Specifically, Our method obtains a relative gain
of over 100% in terms of accuracy on UCF-Crime, compared to TCNN hou2017tube
and C3D. Comparing to C3D, our method boosts the accuracy performance by
obtaining a 13.4% absolute improvement in terms of accuracy.

To analyze the anomalous categories classification performance, we show the
confusion matrix of our method in Fig. 7, where we can observe that the
proposed model can obtain a promising performance of abnormal event
classification. The worst accuracy is from the violence category since the
anomaly samples of the violence category are easily wrongly classified into
the crowd or fighting categories. The best accuracy is obtained from the
crash, falling, fire and fallingtowater categories since their anomaly
definitions are clear, and there is a big gap between these categories and
other categories.

##  6 Conclusion

In this study, we contribute a large-scale benchmark for anomaly detection in
video sequences. It contains 2000 different video sequences with 14 anomaly
categories. We provide annotation data including video-level and frame-level
labels. The proposed database enables research possibility of anomaly
detection in a fully-supervised manner. Then we propose a multi-task
computational model of anomaly detection by effectively learning local and
global spatiotemporal contextual features for video sequences. In the proposed
multi-task deep neural network, the local spatiotemporal features are first
extracted by an Inflated $3$D convolutional network from each video segment.
Then we feed these local spatiotemporal contextual features to a recurrent
convolutional architecture to learn global spatiotemporal contextual features.
Finally, anomaly scores and abnormal event categories are predicted by the
output of the fully convolutional layers of two sub-networks. Comparison
experiments show that the proposed method outperforms the state-of-the-art
anomaly detection methods on public databases and the built LAD database. In
the future, we will further investigate anomaly detection to improve the
performance of anomaly detection for video sequences.

##  7 Acknowledgment

This work was supported in part by the Jiangxi Provincial Natural Science
Foundation under 20202ACB202007, the Fok Ying Tung Education Foundation under
161061, the Foundation of Jiangxi Provincial Department of Education under
Grants 20203BBE53033, the Youth Foundation of Jiangxi Education Department
under Grants GJJ200535 and GJJ190279, and the Postgraduate Innovation Special
Fund of Jiangxi Province, China under Grant YC2020-B139.

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[table]capposition=above othercode=

# On the tree-number of the power graph associated with a finite groups

S. Rahbariyan
Department of Mathematics, Faculty of Sciences,
Arak University
P. O. Box $38156-8-8349$, Arak, Iran
E-mail<EMAIL_ADDRESS>

###### Abstract

Given a group $G$, we define the power graph $\mathcal{P}(G)$ as follows: the
vertices are the elements of $G$ and two vertices $x$ and $y$ are joined by an
edge if $\langle x\rangle\subseteq\langle y\rangle$ or $\langle
y\rangle\subseteq\langle x\rangle$. Obviously the power graph of any group is
always connected, because the identity element of the group is adjacent to all
other vertices. We consider $\kappa(G)$, the number of spanning trees of the
power graph associated with a finite group $G$. In this paper, for a finite
group $G$, first we represent some properties of $\mathcal{P}(G)$, then we are
going to find some divisors of $\kappa(G)$, and finally we prove that the
simple group $A_{6}\cong L_{2}(9)$ is uniquely determined by tree-number of
its power graph among all finite simple groups.

##  1 Introduction

Throughout this paper, all groups are finite and all the graphs under
consideration are finite, simple (with no loops or multiple edges) and
undirected. In this paper, we consider a well-known graph association with a
finite group named as power graph. For a group $G$, the power graph
$\mathcal{P}(G)$, is the graph that its vertices are all elements of the group
$G$ and two different vertices $x$ and $y$ are joined by an edge if $\langle
x\rangle\subseteq\langle y\rangle$ or $\langle y\rangle\subseteq\langle
x\rangle$. We denote by $\mathcal{P}^{*}(G)$ the graph obtained by deleting
the vertex $1$ from $\mathcal{P}(G)$. The term power graph was introduced in
[9], and after that power graphs have been investigated by many authors, see
for instance [1, 5, 14]. The investigation of power graphs associated with
algebraic structures is important, because these graphs have valuable
applications (see the survey article [11]) and are related to automata theory
(see the book [10]).

A spanning tree of a connected graph is a subgraph that contains all the
vertices and is a tree. Counting the number of spanning trees in a connected
graph is a problem of long-standing interest in various field of science. For
a graph $\Gamma$, the number of spanning trees of $\Gamma$; denoted by
$\kappa(\Gamma)$; is known as the complexity of $\Gamma$. By the definition of
the power graph of any group, the identity element of the group is adjacent to
all other vertices, so the graph is always connected. We denote by
$\kappa(G)$, the number of spanning trees of the power graph $\mathcal{P}(G)$
of a group $G$. A well-known result due to Cayley [4] says that the complexity
of the complete graph on $n$ vertices is $n^{n-2}$. In [6] it was shown that a
finite group has a complete power graph if and only if it is a cyclic
$p$-group, where $p$ is a prime number. Thus, as an immediate consequence of
Cayley’s result, we derive $\kappa(\mathbb{Z}_{p}^{m})=(p^{m})^{p^{m}-2}$. In
some investigations, the formula to compute the complexity $\kappa(G)$, for
instance, where $G$ is the cyclic group $\mathbb{Z}_{n}$, dihedral group
$D_{2n}$, the generalized quaternion group $Q_{4n}$ (see [13]), the simple
groups $L_{2}(q)$, the extra-special $p$-groups of order $p^{3}$ and the
Frobenius groups (see [12]) have been obtained. In this paper, we take a step
forward and find some general results due to complexity $\kappa(G)$, for a
finite group $G$, and finally, as an application of these results, we
represent the following investigation.

For two isomorphic groups $G$ and $H$, clearly, $\kappa(G)=\kappa(H)$.
However, generally the converse is not hold. For instance, for all finite
elementary abelian $2$-groups $G$, we have $\kappa(G)=1$.

A group $G$ from a class $\mathcal{C}$ is said to be recognizable in
$\mathcal{C}$ by $\kappa(G)$ (shortly, $\kappa$-recognizable in $\mathcal{C}$)
if every group $H\in\mathcal{C}$ with $\kappa(H)=\kappa(G)$ is isomorphic to
$G$. In other words, $G$ is $\kappa$-recognizable in $\mathcal{C}$ if
$h_{\mathcal{C}}(G)=1$, where $h_{\mathcal{C}}(G)$ is the (possibly infinite)
number of pairwise non-isomorphic groups $H\in\mathcal{C}$ with
$\kappa(H)=\kappa(G)$. We denote by $\mathcal{F}$ and $\mathcal{S}$ the
classes of all finite groups and all finite simple groups, respectively. In
[13], the first example of $\kappa$-recognizable group in class $\mathcal{S}$
was found.

###### Theorem 1.1

[13] The alternating group $A_{5}$ is $\kappa$-recognizable in the class of
all finite simple groups, that is, $h_{\mathcal{S}}(A_{5})=1$.

After that, in [12], the $\kappa$-recognizable group in class $\mathcal{S}$
for simple group $L_{2}(7)$, also, has been proven. However, by the results as
to complexity $\kappa(G)$, which are found in this paper, we are going to
offer a different and short proof for the following theorem.

###### Theorem 1.2

For the simple group $A_{6}\cong L_{2}(9)$, we have
$h_{\mathcal{S}}(A_{6}\cong L_{2}(9))=1$, in the class $\mathcal{S}$ of all
finite simple groups.

##  2 Terminology and Previous Results

The notation and definitions used in this paper are standard and taken mainly
from [2, 3, 15, 17]. We will cite only a few. Let $\Gamma=(V,E)$ be a simple
graph. We denote by $\mathbf{A}=\mathbf{A}(\Gamma)$ the adjacency matrix of
$\Gamma$. The Laplacian matrix $\mathbf{Q}$ of a graph $\Gamma$ is
$\mathbf{\Delta}-\mathbf{A}$, where $\mathbf{\Delta}$ is the diagonal matrix
whose $i$-th diagonal entry is the degree $v_{i}$ in $\Gamma$ and $\mathbf{A}$
is the adjacency matrix of $\Gamma$. The $\mathbf{J}_{m\times n}$ and
$\mathbf{O}_{m\times n}$ denote the matrixes with $m$ rows and $n$ columns,
where each of whose entries is +1, and zero, respectively. Moreover, the
identity matrix is denoted by $\mathbf{I}$. A matrix $\mathcal{A}$ of size
$n$, which is the $n\times n$ square matrix, is denoted by
$\mathcal{A}_{n\times n}$.

When $U\subseteq V$, the induced subgraph $\Gamma[U]$ is the subgraph of
$\Gamma$ whose vertex set is $U$ and whose edges are precisely the edges of
$\Gamma$ which have both ends in $U$. Two graphs are disjoint if they have no
vertex in common, and edge-disjoint if they have no edge in common. If
$\Gamma_{1}$ and $\Gamma_{2}$ are disjoint, we refer to their union as a
disjoint union, and generally denote it by $\Gamma_{1}\oplus\Gamma_{2}$. By
starting with a disjoint union of two graphs $\Gamma_{1}$ and $\Gamma_{2}$ and
adding edges joining every vertex of $\Gamma_{1}$ to every vertex of
$\Gamma_{2}$, one obtains the join of $\Gamma_{1}$ and $\Gamma_{2}$, denoted
$\Gamma_{1}\vee\Gamma_{2}$. A clique in a graph is a set of pairwise adjacent
vertices.

We denote by $\pi(n)$ the set of the prime divisors of a positive integer $n$.
Given a group $G$, we will write $\pi(G)$ instead of $\pi(|G|)$, and denote by
$\pi_{e}(G)$ the set of orders of all elements in a group $G$ and call this
set the spectrum of $G$. The spectrum $\pi_{e}(G)$ of $G$ is closed under
divisibility and determined uniquely from the set $\mu(G)$ of those elements
in $\pi_{e}(G)$ that are maximal under the divisibility relation. In the case
when $\mu(G)$ is a one-element set $\\{n\\}$, we write $\mu(G)=n$. Finally,
two notation $\phi(n)$ and $c_{m}(G)$ are denoted in particular for the
Euler’s totient function, for positive integer $n$, and the number of distinct
cyclic subgroups of order $m$ of $G$. Occasionally, when the group we are
considering is clear from the context, we will simply write $c_{m}$ instead of
$c_{m}(G)$. All further unexplained notation is standard and refers to [3].

At the following, we give several auxiliary results to be used later. First,
we point some important lemmas due to the power graph.

###### Lemma 2.1

[6] Let $G$ be a finite group. Then $\mathcal{P}(G)$ is complete if and only
if $G$ is a cyclic group of order $1$ or $p^{m}$ for some prime number $p$ and
for some natural number $m$.

###### Lemma 2.2

[14] Let $G$ be a finite $p$-group, where $p$ is a prime. Then
$\mathcal{P}^{*}(G)$ is connected if and only if $G$ has a unique minimal
subgroup.

###### Corollary 2.1

[14] Let $G$ be a finite $p$-group, where $p$ is a prime. Then
$\mathcal{P}^{*}(G)$ is connected if and only if $G$ is either cyclic or
generalized quaternion.

###### Lemma 2.3

[14] Let $G$ be a finite group. If $H$ is a subgroup of $G$, then
$\mathcal{P}(H)$ is a subgraph of $\mathcal{P}(G)$. In particular, if $x$ is a
$p$-element of $G$, where $p$ is a prime, then $\langle x\rangle$ is a clique
in $\mathcal{P}(G)$.

In the sequel, we collected some results related to the number of spanning
trees of a simple graph $\Gamma$. The following one is well known, see for
example [[17], Proposition 2.2.8].

###### Theorem 2.1

[17](Deletion-Contraction Theorem) The number of spanning trees of a graph
$\Gamma$ satisfies the deletion-contraction recurrence
$\kappa(\Gamma)=\kappa(\Gamma-e)+\kappa(\Gamma\cdot e)$, where $e\in
E(\Gamma)$. In particular, if $e\in E(\Gamma)$ is a cut-edge, then
$\kappa(\Gamma)=\kappa(\Gamma\cdot e)$.

###### Theorem 2.2

[13] Let $\Gamma$ be a connected graph and let $v$ be a cut vertex of $\Gamma$
with

$\Gamma-v=\Gamma_{1}\oplus\Gamma_{2}\oplus\cdots\oplus\Gamma_{c},$

where $\Gamma_{i}$, $i=1,2,\ldots,c$, is the ith connected component of
$\Gamma-v$ and $c=c(\Gamma-v)$. Set $\tilde{\Gamma_{i}}=\Gamma_{i}+v$. Then,
there holds

$\kappa(\Gamma)=\kappa(\tilde{\Gamma_{1}})\times\kappa(\tilde{\Gamma_{2}})\times\cdots\kappa(\tilde{\Gamma_{c}}).$

###### Lemma 2.4

[13] If $H_{1}$, $H_{2}$, . . . , $H_{t}$ are nontrivial subgroups of a group
$G$ such that $H_{i}\cap H_{j}=\\{1\\}$, for each $1\leqslant i<j\leqslant t$,
then we have $\kappa(G)>\kappa(H_{1})\cdot\kappa(H_{2})\cdots\kappa(H_{t})$.

###### Corollary 2.2

[13] Let $G$ be a finite group and let $p$ be the smallest prime such that
$\kappa(G)<p^{(p-2)}$ Then $\pi(G)\subseteq\pi((p-1)!)$.

###### Theorem 2.3

[16] The number of spanning trees of a graph $\Gamma$ with $n$ vertices is
given by the formula

$\kappa(\Gamma)=\det(\mathbf{J}+\mathbf{Q})/n^{2},$

where $\mathbf{J}$ denotes the matrix each of whose entries is $+1$.

For instance, we consider the power graphs of quaternion group

$Q_{8}=\langle x,y\ |\ x^{4}=1,\ x^{2}=y^{2},\ yx=x^{-1}y\rangle,$

and apply Theorem 2.3, to find $\kappa(Q_{8})$. The power graph
$\mathcal{P}(Q_{8})$ is shown in Fig. 1.

$x^{2}$1$y$$yx^{2}$$yx^{3}$$yx$$x$$x^{3}$Fig. 1. The graph
$\mathcal{P}(Q_{8})$

For this graph, by the definition, the adjacency matrix $\mathbf{A}$ and the
diagonal matrix $\mathbf{\Delta}$ have the following structures:

$\mathbf{A}=\left(\begin{array}[]{c | c | c|c|c}\begin{array}[]{c c}0&1\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&1&1\\\ \begin{array}[]{c c}1&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&1&1\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&1\end{array}&\begin{array}[]{c c}0&0\end{array}&1&1\\\ \begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}1&0\end{array}&\begin{array}[]{c c}0&0\end{array}&1&1\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&1\end{array}&1&1\\\ \begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}1&0\end{array}&1&1\\\ \hline\cr\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&0&1\\\ \hline\cr\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&1&0\\\ \end{array}\right),\ \ \ \ \ \ \ \&\ \ \ \ \ \ \ \mathbf{\Delta}=\left(\begin{array}[]{c | c | c|c|c}\begin{array}[]{c c}3&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&0&0\\\ \begin{array}[]{c c}0&3\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}3&0\end{array}&\begin{array}[]{c c}0&0\end{array}&0&0\\\ \begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&3\end{array}&\begin{array}[]{c c}0&0\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}3&0\end{array}&0&0\\\ \begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&3\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&7&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&0&7\\\ \end{array}\right).$

Therefore

$\mathbf{J}+\mathbf{Q}=\mathbf{J}+(\mathbf{\Delta}-\mathbf{A})=\left(\begin{array}[]{c | c | c|c|c}\begin{array}[]{c c}4&0\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&0&0\\\ \begin{array}[]{c c}0&4\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}4&0\end{array}&\begin{array}[]{c c}1&1\end{array}&0&0\\\ \begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}0&4\end{array}&\begin{array}[]{c c}1&1\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}4&0\end{array}&0&0\\\ \begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}1&1\end{array}&\begin{array}[]{c c}0&4\end{array}&0&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&8&0\\\ \hline\cr\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&\begin{array}[]{c c}0&0\end{array}&0&8\\\ \end{array}\right),$

So by Theorem 2.3 and easy calculation,
$\kappa(Q_{8})=\frac{\det(\mathbf{J}+\mathbf{Q})}{8^{2}}=2^{11}$, as we
expected by the following Theorem.

###### Theorem 2.4

[13] If $n$ is a power of $2$, then the tree-number of the power graph
$\mathcal{P}(Q_{4n})$ is given by the formula $\kappa(Q_{4n})=2^{5n-1}\cdot
n^{2n-2}$.

We conclude this section with two results which are used for our final main
theorem (Theorem 1.2).

###### Theorem 2.5

[7] Let $q=p^{n}$, with $p$ prime and $n\in\mathbb{N}$, let $G=L_{2}(q)$. Then
we have:

$\kappa(G)=p^{\frac{(q^{2}-1)(p-2)}{p-1}}\cdot\kappa(\mathbb{Z}_{\frac{q-1}{k}})^{q(q+1)/2}\cdot\kappa(\mathbb{Z}_{\frac{q+1}{k}})^{q(q-1)/2},$

where $k={\rm gcd}(q-1,2)$, except exactly in the cases $(p,n)=(2,1),(3,1)$.
In particular, we have

$A_{5}\cong L_{2}(5)\cong L_{2}(4)$ and $\kappa(A_{5})=3^{10}\cdot 5^{18}$
(see [13])

$L_{3}(2)\cong L_{2}(7)$ and $\kappa(L_{3}(2))=2^{84}\cdot 3^{28}\cdot 7^{40}$

$A_{6}\cong L_{2}(9)$ and $\kappa(A_{6})=2^{180}\cdot 3^{40}\cdot 5^{108}$.

###### Lemma 2.5

[13] Let $G$ be a finite nonabelian simple group and let $p$ be a prime
dividing the order of $G$. Then $G$ has at least $p^{2}-1$ elements of order
$p$, or equivalently, there is at least $p+1$ cyclic subgroups of order $p$ in
$G$.

##  3 Main parts of manuscrips

As we mentioned before, by the definition of the power graph of any group, the
identity element of the group is adjacent to all other vertices, so the graph
is always connected. In this section, first, we look deeper to the power graph
associated with a finite group and prove some necessarily lemmas, and then we
are going to find some useful divisors of the $\kappa(G)$, for a finite group
$G$.

###### Lemma 3.1

Let $G$ be a $p$-group, where $p$ is a prime. Then $\mathcal{P}^{*}(G)$ has
exactly $c_{p}$ connected component, where $c_{p}$ is the number of distinct
cyclic subgroups of order $p$ of $G$.

Proof. If $\mathcal{P}^{*}(G)$ is connected, then by Lemma 2.2, $G$ has a
unique minimal subgroup ($c_{p}=1$), and so there is nothing to be proved.
Therefore we assume that $\mathcal{P}^{*}(G)$ be disconnected. Obviously, by
the definition of $\mathcal{P}(G)$, in every connected component of
$\mathcal{P}^{*}(G)$, there must be at least one cyclic subgroup of order $p$.
Assume that there are two distinct subgroups $\langle x\rangle$ and $\langle
y\rangle$ in some connected components of $\mathcal{P}^{*}(G)$. Let the
following path, be a shortest path from $x$ to $y$,

$x=x_{0}\sim x_{1}\sim x_{2}\sim\dots\sim x_{n}=y.$

Certainly, $n\geq 2$ and $x\nsim x_{i}$, for each $i=2,3,...,n$. Since $x\sim
x_{1}$, by the definition, we have $x\in\langle x_{1}\rangle$ or
$x_{1}\in\langle x\rangle$. We only consider the first case, and the second
one goes similarly. Since $x_{1}\sim x_{2}$, it follows that $x_{1}\in\langle
x_{2}\rangle$ or $x_{2}\in\langle x_{1}\rangle$. If $x_{1}\in\langle
x_{2}\rangle$, then $x\in\langle x_{2}\rangle$ which is a contradiction.
Therefore, $x_{2}\in\langle x_{1}\rangle$ But then $x,x_{2}\in\langle
x_{1}\rangle$ and since $\langle x_{1}\rangle$ is a $p$-group, Lemma 2.3 shows
that $\langle x_{1}\rangle\setminus\\{1\\}$ is a clique in
$\mathcal{P}^{*}(G)$. Hence $x\sim x_{2}$, a contradiction again. This
completes the proof. $\square$

An immediate consequence of Theorem 2.5 and Lemma 3.1 is the following:

###### Corollary 3.1

Let $G$ be a $p$-group, for some prime numbers $p$, and $H_{i}$ be the ith
connected component of $\mathcal{P}^{*}(G)$. Then

$\kappa(G)=\kappa(\tilde{H_{1}})\times\kappa(\tilde{H_{2}})\times\cdots\times\kappa(\tilde{H_{c_{p}}}),$

where $\tilde{H_{i}}=H_{i}+1$, for $i=1,2,\dots c_{p}$.

Clearly if $H$ is a subgroup of $G$, then $\mathcal{P}^{*}(H)$ is a subgraph
of $\mathcal{P}^{*}(G)$. However, the converse is not true, even if we
consider the vertices of a connected components of $\mathcal{P}^{*}(G)$ in
union with $\\{1\\}$. For instance, by the subgraph
$\mathcal{P}^{*}(\mathbb{Z}_{2}\times\mathbb{Z}_{4})$ (see Fig. 2.), there is
no subgroup $H$ of $G$, in which, $\mathcal{P}^{*}(H)$ be the connected
component with $5$ vertices.

Fig. 2. The subgraph $\mathcal{P}^{*}(\mathbb{Z}_{2}\times\mathbb{Z}_{4})$

At the following, we show that in a particular situation, the converse could
be hold.

###### Lemma 3.2

Let $G$ be a group and $\Omega$ be a connected component of
$\mathcal{P}^{*}(G)$ which is a clique. Then there is a cyclic $p$-subgroup
$H$ of $G$ with $p\in\pi(G)$ that $\mathcal{P}^{*}(H)=\Omega$.

Proof. By the definition of power graph, if $(o(g_{1}),o(g_{2}))=1$, for
$g_{1},g_{2}\in G$, then $g_{1}\nsim g_{2}$ in $\mathcal{P}(G)$. Therefore,
since $\Omega$ is a clique, for every element $\alpha$ in $V(\Omega)$, we must
have $\pi(o(\alpha))=p$, for a prime number $p$. Let $\beta\in V(\Omega)$ be
the vertex which its order has the largest power of the prime p. We claime
that $\Omega=\mathcal{P}^{*}(\langle\beta\rangle)$. For every element $\alpha$
in $V(\Omega)$, if $\alpha\in\langle\beta\rangle$, then there is nothing to be
proved. Let $\beta\in\langle\alpha\rangle$, but then $o(\beta)\ |\ o(\alpha)$,
which implies that $o(\alpha)=o(\beta)$, because $o(\beta)$ is the largest
power of the prime p, and so $\langle\alpha\rangle=\langle\beta\rangle$. This
proves our claim. $\square$

###### Corollary 3.2

Let $G$ be a $p$-group, for prime number $p$. If all connected components of
$\mathcal{P}^{*}(G)$ are clique, then $\pi(\kappa(G))=\\{p\\}$.

Proof. The proof is straightforward, by Lemma 3.2 and Theorem 2.5. $\square$

Now, we are ready to represent our results due to some useful divisors of the
$\kappa(G)$, for a finite group $G$.

###### Lemma 3.3

Let $G$ be a group and $p\in\mu(G)$, for some prime number $p$ in $\pi(G)$,
then $p^{p-2}\ |\ \kappa(G)$.

Proof. Let $g\in G$ be an element of order $p$, for a prime number
$p\in\mu(G)$. By Lemma 2.1, $\mathcal{P}(\langle g\rangle)$ is a complete
graph. Hence, we only need to prove $\langle g\rangle$ is a connected
component in $\mathcal{P}^{*}(G)$. Assume that $g$ and $h$ are adjacent, for
some $h\in G$. If $h\in\langle g\rangle$, then there is nothing to be proved.
Let $g\in\langle h\rangle$ or equivalently $g=h^{\alpha}$, for some $\alpha$.
But then $o(g)\ |\ o(h)$, which implies that $o(g)=o(h)$ because
$o(g)\in\mu(G)$, and so $\langle g\rangle=\langle h\rangle$. Therefor,
$\langle g\rangle$ is a connected component in $\mathcal{P}^{*}(G)$, and so by
Theorem 2.5, $\kappa(\langle g\rangle)=p^{p-2}\ |\ \kappa(G)$, as required.
$\square$

The following result is not limited to only connected algebraic graphs, and
holds for all connected simple graphs.

###### Lemma 3.4

Let $\Gamma$ be a simple graph and $\\{v_{1},v_{2},\ldots,v_{k}\\}\subseteq
V(\Gamma)$. If the vertices $v_{1},v_{2},\ldots,v_{k}$ have full-degree, then
$|V(\Gamma)|^{k}\ |\ \det((\mathbf{J}+\mathbf{Q})$.

Proof. Let $|V(\Gamma)|=n$. Since $v_{1},v_{2},\ldots,v_{k}$ have full-degree,
the adjacency matrix $\mathbf{A}(\Gamma)$ and the diagonal matrix
$\mathbf{\Delta}(\Gamma)$ have the following structures:

$\mathbf{A}(\Gamma)=\left(\begin{array}[]{c | c}\mathbf{J}_{k\times k}-\mathbf{I}_{k\times k}&\mathbf{J}_{k\times(n-k)}\\\ &\\\ \hline\cr&\\\ \mathbf{J}_{(n-k)\times k}&\mathbf{A}(\Gamma\setminus\\{v_{1},\ldots v_{k}\\})\\\ \end{array}\right),\ \ \ \&\ \ \ \ \mathbf{\Delta}(\Gamma)=\left(\begin{array}[]{c | c}(n-1)\mathbf{I}_{k\times k}&\mathbf{O}_{k\times(n-k)}\\\ &\\\ \hline\cr&\\\ \mathbf{O}_{(n-k)\times k}&\Theta_{(n-k)\times(n-k)}\\\ \end{array}\right),$

where $\Theta_{(n-k)\times(n-k)}$ is the diagonal matrix whose diagonal
entries are the degree of vertexes $V(\Gamma)\setminus\\{v_{1},\ldots
v_{k}\\}$ in $\Gamma$. Therefore the matrix
$\mathbf{J}+\mathbf{Q}=\mathbf{J}+(\mathbf{\Delta}(\Gamma)-\mathbf{A}(\Gamma))$
has the following structure:

$\mathbf{J}+\mathbf{Q}=\left(\begin{array}[]{c | c}n\mathbf{I}_{k\times k}&\mathbf{O}_{k\times(n-k)}\\\ &\\\ \hline\cr&\\\ \mathbf{O}_{(n-k)\times k}&\mathbf{J}_{(n-k)\times(n-k)}-(\mathbf{A}(\Gamma\setminus\\{v_{1},\ldots v_{k}\\})-\Theta_{(n-k)\times(n-k)})\\\ \end{array}\right).$

Now, by the structure of $\mathbf{J}+\mathbf{Q}$, we have

$\det(\mathbf{J}+\mathbf{Q})=\det(n\mathbf{I}_{k\times
k})\cdot\det(\mathbf{J}_{(n-k)\times(n-k)}-(\mathbf{A}(\Gamma\setminus\\{v_{1},\ldots
v_{k}\\})-\Theta_{(n-k)\times(n-k)})),$

and so $n^{k}\ |\ \det(\mathbf{J}+\mathbf{Q})$, as a required. $\square$

In Lemma 3.3, we find a divisor for $\kappa(G)$, when a prime divisor of $G$
be in the $\mu(G)$. In the next theorem, we extend the result for any number
in the $\mu(G)$, and represent a divisor for
$\det(\mathbf{J}+\mathbf{Q}(\mathcal{P}(G))$.

###### Theorem 3.1

Let $G$ be a finite group and $m\in\mu(G)$. Then $|G|\cdot m^{\phi(m)}\ |\
\det(\mathbf{J}+\mathbf{Q}(\mathcal{P}(G))$.

Proof. Let $|G|=n$. By Lemma 3.4, since the vertex $1$ has full-degree, we
have

$\det(\mathbf{J}_{n\times
n}+\mathbf{Q}(\mathcal{P}(G)))=n\cdot\det(\mathbf{J}_{(n-1)\times(n-1)}+(\mathbf{\Delta}-\mathbf{A}(\mathcal{P}^{*}(G)))),$

where $\mathbf{\Delta}$ is the diagonal matrix whose diagonal entries are the
degree of vertexes in $\mathcal{P}^{*}(G)$ in $\mathcal{P}(G)$.

Suppose that $g\in G$ be an element of order $m$. By the definition of power
graph, since $x\in G$ is adjacent to $g$ if and only if $\langle
x\rangle\leq\langle g\rangle$, therefore $\mathbf{A}(\mathcal{P}^{*}(G))$ and
$\mathbf{\Delta}$ have the following block-matrix structure:

$\mathbf{A}(\mathcal{P}^{*}(G))=\left(\begin{array}[]{c | c c c}(\mathbf{J}-\mathbf{I})_{\phi(m)\times\phi(m)}&\mathbf{J}_{\phi(m)\times((m-1)-\phi(m))}\ \ \ \ \ \vline&\mathbf{O}_{\phi(m)\times(n-m)}\\\ &&&\\\ \hline\cr&&&\\\ \mathbf{J}_{((m-1)-\phi(m))\times\phi(m)}&&\\\ &&&\\\ \cline{1-1}\cr&\ \ \ \ \ \ \ \ \ \ \ \ \Omega_{((n-1)-\phi(m))\times((n-1)-\phi(m))}&\\\ \mathbf{O}_{(n-m)\times\phi(m)}&&\end{array}\right),$

where
$\Omega_{((n-1)-\phi(m))\times((n-1)-\phi(m))}=\mathbf{A}(\mathcal{P}(G)\setminus\langle
g\rangle)$, and

$\mathbf{\Delta}=\left(\begin{array}[]{c | c c c}(m-1)\mathbf{I}_{\phi(m)\times\phi(m)}&\mathbf{O}_{\phi(m)\times((n-1)-\phi(m))}\\\ &&&\\\ \hline\cr&&&\\\ \mathbf{O}_{((n-1)-\phi(m))\times\phi(m)}&\ \ \ \ \ \ \ \ \ \ \ \ \Lambda_{((n-1)-\phi(m))\times((n-1)-\phi(m))}&\\\ \end{array}\right),$

where $\Lambda_{((n-1)-\phi(m))\times((n-1)-\phi(m))}$ is the diagonal matrix
whose diagonal entries are the degree of elements of
$\mathcal{P}(G)\setminus\langle g\rangle$ in $\mathcal{P}(G)$.

On the other hand, by considering the Laplacian matrix
$\mathbf{Q}=\mathbf{\Delta}-\mathbf{A}(\mathcal{P}^{*}(G)))$, we must have the
following block-matrix structure for
$\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q}$:

$\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q}=\left(\begin{array}[]{c | c c c}m\mathbf{I}_{\phi(m)\times\phi(m)}&\mathbf{O}_{\phi(m)\times((m-1)-\phi(m))}\ \ \ \ \ \vline&\mathbf{J}_{\phi(m)\times(n-m)}\\\ &&&\\\ \hline\cr&&&\\\ \mathbf{O}_{((m-1)-\phi(m))\times\phi(m)}&&\\\ &&&\\\ \cline{1-1}\cr&\ \ \ \ \ \ \ \ \ \ \ \ \Theta_{((n-1)-\phi(m))\times((n-1)-\phi(m))}&\\\ \mathbf{J}_{(n-m)\times\phi(m)}&&\end{array}\right),$

where

${\Theta}_{((n-1)-\phi(m))\times((n-1)-\phi(m))}=\mathbf{J}_{{((n-1)-\phi(m))\times((n-1)-\phi(m))}}+\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ $ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
(\Lambda_{((n-1)-\phi(m))\times((n-1)-\phi(m))}-\Omega_{((n-1)-\phi(m))\times((n-1)-\phi(m))}).$

In the sequel, $Ri$ and $Cj$ respectively designate the row $i$ and the column
$j$ of the matrix $\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q}$. We apply the
following row and column operations in the
$\det(\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q})$. We subtract row $R_{1}$ from
row $R_{i}$, for $i=2,3,\ldots,\phi(m)$ and subsequently we add column $C_{j}$
to column $C_{1}$, for $j=2,3,\ldots,\phi(m)$. It is not too difficult to see
that, step by step, we have:

$\det(\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q})=\det(\left(\begin{array}[]{c | c c c}m\mathbf{I}_{\phi(m)\times\phi(m)}&\mathbf{O}_{\phi(m)\times((m-1)-\phi(m))}\ \ \ \ \ \vline&\begin{array}[]{ccc}1&\dots&1\\\ 0&\dots&0\\\ \vdots&\vdots&\vdots\\\ 0&\dots&0\end{array}\\\ &&&\\\ \hline\cr&&&\\\ \mathbf{O}_{((m-1)-\phi(m))\times\phi(m))}&&\\\ &&&\\\ \cline{1-1}\cr&\ \ \ \ \ \ \ \ \ \ \ \ \Theta_{((n-1)-\phi(m))\times((n-1)-\phi(m))}&\\\ \begin{array}[]{cccc}\phi(m)&1&\dots&1\\\ \vdots&\vdots&\dots&\vdots\\\ \phi(m)&1&\dots&1\end{array}\\\ &&\end{array}\right)),$

again, we subtract $\frac{1}{m}C_{1}$ from $C_{j}$, for $j=m,\ldots,n-1$, and
so

$\det(\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q})=\det(\left(\begin{array}[]{c | c c c}m\mathbf{I}_{\phi(m)\times\phi(m)}&\mathbf{O}_{((n-1)-\phi(m))\times((n-1)-\phi(m))}&\\\ &&&\\\ \hline\cr&&&\\\ \mathbf{O}_{((m-1)-\phi(m))\times\phi(m)}&&\\\ &&&\\\ \cline{1-1}\cr&\ \ \ \ \ \ \ \ \ \ \ \ \Theta_{((n-1)-\phi(m))\times((n-1)-\phi(m))}-\Upsilon\\\ \begin{array}[]{cccc}\phi(m)&1&\dots&1\\\ \vdots&\vdots&\dots&\vdots\\\ \phi(m)&1&\dots&1\end{array}\\\ &&\end{array}\right)),$

where $\Upsilon$ has the following structure

$\Upsilon=\left(\begin{array}[]{c|c }&1\ \ \ \ \ \ldots\ \ \ \ \ 1\\\ &\\\
\cline{2-2}\cr\mathbf{O}_{((n-1)-\phi(m))\times((m-1)-\phi(m))}&\\\ &\\\
&\mathbf{O}_{((m-1)-\phi(m))\times(n-m)}\\\ &\\\ \cline{2-2}\cr&\\\
&\frac{\phi(m)}{m}\mathbf{J}_{(n-m)\times((n-m)}\\\ \end{array}\right),$

and so

$\det(\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q})=m^{\phi(m)}\cdot\det(\Theta_{((n-1)-\phi(m))\times((n-1)-\phi(m))}-\Upsilon).$

On the other hand, by above mention discussion, $\det(\mathbf{J}_{n\times
n}+\mathbf{Q}(\mathcal{P}(G)))=n\cdot\det(\mathbf{J}_{(n-1)\times(n-1)}+\mathbf{Q}$.
Therefore

$n\cdot m^{\phi(m)}\ |\ \det(\mathbf{J}_{n\times
n}+\mathbf{Q}(\mathcal{P}(G))),$

as we claimed. $\square$

###### Corollary 3.3

Let $G$ be a group and $g\in G$. If the degree of $g$ is $k$ in
$\mathcal{P}(G)$, then

$|G|\cdot(k+1)^{\phi(|g|)}\ |\ \det(\mathbf{J}_{n\times
n}+\mathbf{Q}(\mathcal{P}(G))).$

Proof. By the exact same way in the proof of Theorem 3.2, the result is
straightforward. $\square$

At the end, as an application of our main results, we are going to prove
Theorem 1.2. As a matter of fact, we verify that $h_{\mathcal{S}}(A_{6}\cong
L_{2}(9))=1$, in the class $\mathcal{S}$ of all finite simple groups.

Proof. For the proving of Theorem 1.2, assume that $G\in\mathcal{S}$, with
$\kappa(G)=\kappa(A_{6})=2^{180}\cdot 3^{40}\cdot 5^{108}$ (see Theorem 2.5).
First of all, $G$ is a non-abelian simple group. Otherwise,
$\kappa(G)=\kappa(\mathbb{Z}_{p})=p^{p-2}$, for some prim number $p$, which is
a contradiction.

In the next, we claim that $\pi(G)\subseteq\\{2,3,5,7,11\\}$. By Lemma 2.5, we
have $c_{p}\geqslant p+1$, where $p\in\pi(G)$ and $c_{p}$ is the number of
cyclic subgroups of order $p$ in $G$. Therefore, by Lemma 2.4,

$13^{11\cdot 14}\gneqq 2^{180}\cdot 3^{40}\cdot
5^{108}=\kappa(G)>\kappa({\mathbb{Z}}_{p})^{c_{p}}\geqslant\kappa({\mathbb{Z}}_{p})^{p+1}=p^{(p-2)(p+1)},$

which leads us to the conclusion.

Finally we are going to show that $G\cong A_{6}\cong L_{2}(9)$. If $7$ or $11$
be an element in $\mu(G)$, then by Lemma 3.3, $7^{5}$, or $11^{9}$ divide
$\kappa(G)$, which is a contradiction. Now, by results collected in [18], $G$
is isomorphic to one of the groups $A_{5}\cong L_{2}(4)\cong L_{2}(5)$,
$A_{6}\cong L_{2}(9)$, $S_{4}(7)$. By Theorem 1.1, $G\ncong A_{5}$. If $G\cong
S_{4}(7)$, then by Theorem 3.1, since $56\in\mu(G)$,

$|G|\cdot(56)^{\phi}(56)=|S_{4}(7)|\cdot 2^{72}\cdot 7^{24}\ |\
\det(\mathbf{J}+\mathbf{Q}),$

which concludes that (by Theorem 2.3) $7^{20}\ |\ \kappa(G)$, again we have a
contradiction. Therefore $G\cong A_{6}$, and the proof has been completed.
$\square$

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