# Topological and spectral properties of random digraphs

C. T. Martínez-Martínez,1 J. A. Méndez-Bermúdez,2 and José M. Sigarreta1
1Universidad Autónoma de Guerrero, Centro Acapulco CP 39610, Acapulco de
Juárez, Guerrero, Mexico
2Instituto de Física, Benemérita Universidad Autónoma de Puebla, Puebla 72570,
Mexico

###### Abstract

We investigate some topological and spectral properties of Erdős-Rényi (ER)
random digraphs $D(n,p)$. In terms of topological properties, our primary
focus lies in analyzing the number of non-isolated vertices $V_{x}(D)$ as well
as two vertex-degree-based topological indices: the Randić index $R(D)$ and
sum-connectivity index $\chi(D)$. First, by performing a scaling analysis we
show that the average degree $\left\langle k\right\rangle$ serves as scaling
parameter for the average values of $V_{x}(D)$, $R(D)$ and $\chi(D)$. Then, we
also state expressions relating the number of arcs, spectral radius, and
closed walks of length 2 to $(n,p)$, the parameters of ER random digraphs.
Concerning spectral properties, we compute six different graph energies on
$D(n,p)$. We start by validating $\left\langle k\right\rangle$ as the scaling
parameter of the graph energies. Additionally, we reformulate a set of bounds
previously reported in the literature for these energies as a function
$(n,p)$. Finally, we phenomenologically state relations between energies that
allow us to extend previously known bounds.

##  I Introduction

In recent years, there has been a significant increase in the use of graphs to
represent complex systems in various fields, including computer science,
engineering, biology, and social sciences B16 ; N10 ; BA99 ; AB02 ; GMDMSP18 .
This growing trend can be attributed to the effectiveness of capturing the
properties of complex systems through graphs, where the vertices represent the
agents of the system and the edges reflect their interactions. This, in turn,
opens the door to the analysis of complex systems through various mathematical
techniques coming mainly from graph theory.

The study of the properties of graphs covers many aspects, focusing mainly on
topological and spectral properties. One of the ways to study and characterize
these properties is through their topological descriptors, such as degree
distribution, clustering coefficient, eigenvector centrality, energy, and,
more recently, topological indices N03 ; LM01 ; NMHPLB18 ; MZN14 .

Although many studies have been carried out with highly relevant results about
the topological and spectral properties of graphs, most of them focus on
graphs whose edges do not have a specific direction, in which the connection
between two vertices is symmetric and bidirectional (undirected graphs).
However, in several cases, it is mandatory to incorporate the direction of the
information flow when considering the modeling of real-world systems. This is
indeed the case when considering food webs ABB05 ; L80 ; ZU03 , neural
networks SZ04 ; SCKH04 ; RS10 , genetic regulation BGL11 , chemical networks
VRJ13 ; PDSC12 , fluid flows SMCN10 ; K021  or financial networks AOT15 ,
among many other relevant applications. In these scenarios, it is crucial to
capture the orientation of the connections so that the systems are represented
by directed networks, commonly known as digraphs. Consequently, there is a
specific interest in exploring the properties of directed graphs.

A digraph or directed graph is a mathematical structure denoted as $D=(V,E)$,
where $V$ represents a finite set of $n$ elements called vertices or nodes and
$E\subset V\times V$ comprises $m$ directed edges (also called arcs)
connecting vertices.

Topological properties delve into the fundamental structural properties of
digraphs, including connectivity, accessibility, cycles, and paths. In this
line, applying topological indices based on vertex degrees to characterize and
analyze the topological properties of graphs has been a widely used approach.
The concept of an index based on vertex degrees originates in chemical graph
theory, which uses graph theory to study the properties of chemical compounds
by representing them as graphs, where atoms are vertices and bonds are edges.
The vertex-degree-based (VDB) topological indices quantify some aspects of the
topology of the graph in relation to the degrees of its vertices. In a general
formulation, a VDB topological index can be expressed as G13 :

$TI=TI(G)=\sum_{i\sim j}f(k_{i},k_{j}),$ (1)

where the summation extends over all pairs of adjacent vertices, denoted as
$i$ and $j$, within the molecular graph $G$, $k_{i}$ is the degree of the
vertex $i$ and $f(k_{i},k_{j})$ represents a function tailored to the specific
topological property under investigation. Since applying these indices to the
study and characterization of the topological properties of graphs has
acquired great relevance, many topological indices have been proposed.
However, extending this concept to directed graphs is a complex task since, in
digraphs, each vertex has an out-degree, an in-degree, and a total degree.
However, Monsalve and Rada have recently presented a generalization of VDB
topological indices applied to digraphs MR21 . Consequently, there are still
few works in which the properties of these topological indices have been
explored MR21b ; CMR22 ; AA22 .

On the other hand, the study of spectral properties involves the study of
eigenvalues and eigenvectors associated with matrices corresponding to
digraphs, such as the adjacency matrix, the Laplacian matrix, and the
Hermitian matrix, among others. In this context, also rooted in chemical graph
theory, the concept of energy emerges as a spectral quantity that serves as a
descriptor of the properties of a graph and allows the characterization and
study of the properties of specific systems. The concept of energy was
initially introduced using the eigenvalues of the adjacency matrix associated
with a graph: For a simple undirected graph, the adjacency matrix is defined
through the matrix elements

$A_{ij}=\left\\{\begin{array}[]{ll}1&\text{ if there is an edge between
vertices $i$ and $j$, }\\\ 0&\text{ otherwise. }\end{array}\right.$ (2)

In 1978 Ivan Gutman proposed the concept of energy of a finite and undirected
simple graph based on Huckel’s orbital model as G78 ; G01

$E(G)=\sum_{i=1}^{n}|\lambda_{i}|,$ (3)

where $\lambda_{i}$ are the eigenvalues of the adjacency matrix of the graph.
Furthermore, other energies associated with other graph matrices have been
proposed, such as the Laplacian energy GZ06 , the Laplacian-energy like
invariant LL08 , the signless Laplacian energy SRAG , the distance energy
IGV10 , the incidence energy JKM09 , the skew energy ABS10 , the Sombor energy
GN21 , the Randić energy BGGC10 , the Seidel energy H12 , etc.

Moreover, the Coulson integral is a complex integral that allows computing the
energy of a graph without directly calculating its eigenvalues: Let $\phi$ be
the characteristic polynomial of the adjacency matrix of the graph, then
$\phi$ is the characteristic polynomial of the graph, which is defined as

$\phi(G,x)=\det[xI-A(G)],$ (4)

where $I$ is the identity matrix of order $n$. The Coulson integral is defined
as C40

$E(G)=\frac{1}{\pi}\int_{-\infty}^{\infty}\left(n-\frac{ix\phi^{\prime}(G,ix)}{\phi(G,ix)}\right)dx,$
(5)

where $\phi^{\prime}(G,ix)$ is the derivative of $\phi(G,x)$ and $n$ is the
order of the adjacency matrix.

The energy of a graph has several applications in various fields, such as
chemistry, physics, mathematics, biology, social networks, computer science,
etc DME13 ; DGF12 ; XXZCG20 ; SSN17 ; KM19 . It is mainly used as an indicator
of the graph structure that determines specific properties of the system
represented by the graph or to optimize specific processes. Furthermore, the
graph energy has been used as a criterion for graph classification. Depending
on the value of their energy, graphs can be categorized as hyperenergetic if
$E(G)>2(n-1)$ or non-hyperenergetic if $E(G)\leq 2(n-1)$; the energy value of
a complete graph serves as a reference in this sense B04 .

Thus, the interest in the study of the energy of a graph has grown
significantly. The most notable results in this field focus mainly on
determining upper and lower bounds for this magnitude based on various
properties of the graphs, mainly of a topological nature. One of the most
important bounds on the energy of a graph is the McClelland inequality, which
establishes a relationship between the energy and the number of vertices and
edges of the corresponding graph M71 :

$E(G)\leq\sqrt{2mn}.$ (6)

It is important to notice that for digraphs, the adjacency matrix is not
necessarily symmetric, so its eigenvalues can be complex, and the definition
of the graph energy of Eq. (3) cannot be straightforwardly extended. Given
this, several definitions of digraph energies have also been proposed and
studied. Therefore, this work investigates topological and spectral
characteristics of directed random graphs, focusing on the Erdős-Rényi model.

##  II Topological and spectral properties of Erdős-Rényi digraphs

An Erdős-Rényi (ER) digraph, denoted by $D(n,p)$, is a directed random graph
with $n$ independent vertices connected with probability $p$. Given two
vertices $u$ and $v$, $p$ is the probability that there is an arc from vertex
$u$ to vertex $v$, so $p\in(0,1)$. When $p=0$, the graph consists of $n$
isolated vertices; when $p=1$, it becomes a complete graph. We can obtain
graphs between these two extremes by varying the value of $p$ between $0$ and
$1$. It is important to note that for $0<p<1$, a given pair of parameters
$(n,p)$ represents an infinity set of random graphs. Therefore, calculating a
property for a single graph is not informative. Instead, we can obtain more
relevant information by calculating a given average property over an ensemble
of random graphs characterized by the same pair of parameters $(n,p)$.
Although this statistical approach is a common practice in random matrix
theory (RMT), it is not as common in graph theory; however, it has been
applied recently to several random graph models MMRS20 ; AHMS20 ; AMRS20 ;
MMRS21 ; MAMRP15 ; PRRCM20 ; PM23 .

Thus, below, we perform a numerical analysis of some topological properties of
ER digraphs by the use of the number of non-isolated vertices ($V_{x}(D)$) and
the Randić ($R(D)$) and the sum-connectivity ($\chi(D)$) indices.

Following the generalization of the concept of VDB topological indices of
digraphs proposed by Monsalve and Rada MR21 , the Randić and the sum-
connectivity indices are respectively defined as:

$R(D)=\frac{1}{2}\sum_{uv\in D}\frac{1}{\sqrt{{k_{u}}^{+}{k_{v}}^{-}}}$ (7)

and

$\chi(D)=\frac{1}{2}\sum_{uv\in D}\frac{1}{\sqrt{{k_{u}}^{+}+{k_{v}}^{-}}},$
(8)

where $uv$ denotes the arc connecting vertices $u$ and $v$, ${k_{u}}^{+}$
denotes the out-degree of the vertex $u$, and ${k_{v}}^{-}$ denotes the in-
degree of the vertex $v$.

First, we compute the average values of $V_{x}(D)$, $R(D)$ and $\chi(D)$ for
ensembles of adjacency matrices of ER digraphs characterized by different
combinations of parameters $(n,p)$. In Fig. 1, these quantities are shown for
four different graph sizes as a function of the connection probability $p$. We
can observe that the curves corresponding to each quantity follow similar
shapes but are displaced in the $p$-axis depending on the graph size. To
better appreciate the shape of these curves, we normalize them to the size of
the network and plot them again in Fig. 2.

Figure 1: (a) Average number of non-isolated vertices $\left\langle
V_{x}(D)\right\rangle$, (b) average Randić index $\left\langle
R(D)\right\rangle$ and (c) average sum-connectivity index
$\left\langle\chi(D)\right\rangle$ as a function of the connection probability
$p$ of Erdős-Rényi digraphs of different sizes $n\in[50,400]$. Each symbol was
calculated by averaging over $10^{6}/n$ random digraphs $D(n,p)$. Figure 2:
(a) $\left\langle V_{x}(D)\right\rangle$, (b) $\left\langle R(D)\right\rangle$
and (c) $\left\langle\chi(D)\right\rangle$ normalized to $n$ as a function of
the connection probability $p$ of Erdős-Rényi digraphs of different sizes
$n\in[50,400]$. Dotted lines in panels (a-c) correspond to $0.5$, $0.25$ and
$0.1$, respectively. Same data sets of Fig. 1.

In Figs. 2(a) and (b), it can be seen that the shape of the normalized curves
of $\left\langle V_{x}(D)\right\rangle$ and $\left\langle R(D)\right\rangle$
are very similar. Initially, for small values of $p$, both are close to zero,
then they increase with $p$ until they reach their maximum values. In the case
of $\left\langle V_{x}(D)\right\rangle/n$, the maximum value is $1$, while for
$\left\langle R(D)\right\rangle/n$ it is $1/2$. However, Fig. 2(c) shows a
different picture for the normalized curves of
$\left\langle\chi(D)\right\rangle$. That is,
$\left\langle\chi(D)\right\rangle/n$ is a strictly monotone increasing
function and its maximum value depends on the graph size. The maximum value of
$\left\langle\chi(D)\right\rangle/n$ is reached at $p=1$ and is equal to
$\sqrt{(n-1)/8}$.

Notably, in all three cases, the curves corresponding to the same quantity
exhibit a similar behavior but they are shifted along the $p$-axis for
different graph sizes $n$. Now, our goal is to identify a scaling parameter
for these quantities. To achieve this, we first need to quantify the
displacement of the curves with $n$. Then, without loss of generality, we
characterize the displacement by computing the value of $p$ (that we label as
$p^{*}$) for which $\left\langle V_{x}(D)\right\rangle/n$, $\left\langle
R(D)\right\rangle/n$ and $\left\langle\chi(D)\right\rangle/n$ reach the value
of 0.5, 0.25 and 0.1, respectively; see the dotted lines in Fig. 2.

Figure 3: $p^{*}$ for (a) $\left\langle V_{x}(D)\right\rangle$, (b)
$\left\langle R(D)\right\rangle$ and (c) $\left\langle\chi(D)\right\rangle$ as
a function of the graph size $n$ of Erdős-Rényi digraphs.

In Fig. 3 we present $p^{*}$ as a function of the graph size $n$ and observe a
linear trend of the data sets $p^{*}$ vs. $n$ (in log-log scale), suggesting a
power–law behavior of the form

$p^{*}=\mathcal{C}n^{-\beta}.$ (9)

Then, by performing numerical fittings, we determined the parameters
$\mathcal{C}$ and $\beta$ which are reported in Table 1. There, we can clearly
see that $\beta\approx 1$ in all three cases. Hence, we define the scaling
parameter $\xi$ as the ratio $p/p^{*}$,

$\xi=\frac{p}{p^{*}}\propto\frac{p}{n^{\beta}}\propto\frac{p}{n^{-1}}=np.$
(10)

Table 1: Values of the constants $\mathcal{C}$ and $\beta$ obtained by fittings of Eq. (9) to the data in Fig. 3. | $\left\langle V_{x}(D)\right\rangle$ | $\left\langle R(D)\right\rangle$ | $\left\langle\chi(D)\right\rangle$
---|---|---|---
$\mathcal{C}$ | $0.3612$ | $0.7608$ | $0.3271$
$\beta$ | $1.0051$ | $1.0026$ | $1.005$

Previous studies on undirected ER graphs have demonstrated that topological
measures can be scaled with the average degree $\left\langle k\right\rangle$
MAMRP15 ; MMRS20 ; MMRS21 . Here, the average degree of ER digraphs is given
by

$\left\langle k\right\rangle=2(n-1)p.$ (11)

In addition, we can observe that both $\left\langle k\right\rangle$ and $\xi$
depend on $n$ and $p$ in the same functional form. Therefore, we can express
$\xi$ as a function of $\left\langle k\right\rangle$ and vice versa. Also, it
is important to recall that the scaling parameter is not unique; a function of
it can also serve as scaling parameter. These observations allow us to propose
the average degree $\left\langle k\right\rangle$ as the scaling parameter for
the topological properties of ER digraphs. Then, in Fig. 4 we present again
the curves of Fig. 2 but now plotted as a function of $\left\langle
k\right\rangle$. As observed, the average degree indeed serves as the scaling
parameter of these topological quantities.

Figure 4: (a) $\left\langle V_{x}(D)\right\rangle$, (b) $\left\langle
R(D)\right\rangle$ and (c) $\left\langle\chi(D)\right\rangle$ normalized to
$n$ as a function of the average degree $\left\langle k\right\rangle$ of
Erdős-Rényi digraphs of different sizes $n\in[50,400]$. Same data sets of Fig.
1.

Other important quantities in the study of digraphs are the number of arcs
$m$, the number of closed walks of length two $c_{2}$, and the spectral radius
$\rho$ (the maximum of the absolute values of the adjacency matrix
eigenvalues). Our next goal is to compute these quantities and examine whether
they can also be scaled with the average degree. To achieve this, we construct
ensembles of ER digraphs characterized by different combinations of parameters
and compute the average of the quantities above. In Fig. 5 we plot
$\left\langle m\right\rangle$, $\left\langle c_{2}\right\rangle$ and
$\left\langle\rho\right\rangle$ as a function of the connection probability
$p$. Remarkably, these quantities exhibit a behavior similar to that reported
for the previously studied topological indices: Curves representing the same
quantity show a similar pattern but they are shifted along the $p$-axis for
different graph sizes. This observation strongly suggests that these
quantities may also be scaled with the average degree. Furthermore, Fig. 5
reveals noteworthy characteristics. Specifically, in the case of $\left\langle
m\right\rangle$ and $\left\langle c_{2}\right\rangle$, we observe a linear
trend with $p$ on a log-log scale. Numerical calculations indicate that
$\left\langle m\right\rangle$ follows the relationship $\left\langle
m\right\rangle\approx n^{2}p$. Similarly, for $\left\langle
c_{2}\right\rangle$ we find that $\left\langle c_{2}\right\rangle\approx
n^{2}p^{2}/2$. Additionally, for $p>0.01$, we find that
$\left\langle\rho\right\rangle\approx np$. These approximations, where the
average degree can be easily identified (i.e. $np\approx\left\langle
k\right\rangle/2$), are indicated in each panel of Fig. 5 with dashed lines.

Figure 5: (a) Average number of edges $\left\langle m\right\rangle$, (b)
average number of closed walks of length 2 $\left\langle c_{2}\right\rangle$
and (c) average spectral radius $\left\langle\rho\right\rangle$ as a function
of the connection probability $p$ of Erdős-Rényi digraphs of different sizes
$n\in[50,400]$. Dashed lines in panels (a-c) correspond to $\left\langle
m\right\rangle=n^{2}p$, $\left\langle c_{2}\right\rangle=n^{2}p^{2}/2$, and
$\left\langle\rho\right\rangle=np$, respectively. Each symbol was calculated
by averaging over $10^{6}/n$ random digraphs. Figure 6: (a) $\left\langle
m\right\rangle/n$, (b) $\left\langle c_{2}\right\rangle$ and (c)
$\left\langle\rho\right\rangle$ as a function of the average degree
$\left\langle k\right\rangle$ of Erdős-Rényi digraphs of different sizes
$n\in[50,400]$. Dashed lines in panels (a-c) correspond to $\left\langle
m\right\rangle/n=\left\langle k\right\rangle/2$ , $\left\langle
c_{2}\right\rangle=\left\langle k\right\rangle^{2}/8=m^{2}/2n^{2}$ and
$\left\langle\rho\right\rangle=\left\langle k\right\rangle/2$ , respectively.
Same data sets of Fig. 5.

Then, in Fig. 6 we verify that the average degree indeed scales $\left\langle
m\right\rangle/n$, $\left\langle c_{2}\right\rangle$ and
$\left\langle\rho\right\rangle$. Therefore we can finally write $\left\langle
m\right\rangle/n\approx\left\langle k\right\rangle/2$, $\left\langle
c_{2}\right\rangle\approx\left\langle k\right\rangle^{2}/8\approx
m^{2}/2n^{2}$ and $\left\langle\rho\right\rangle\approx\left\langle
k\right\rangle/2$ for $k>1$; see the dashed lines in the corresponding panels
of Fig. 6. These findings provide highly relevant information about the
relationships and scaling behavior of topological quantities, such as $m$ and
$c_{2}$, and the spectral measure $\rho$, in relation to the graph parameters
$p$, $n$ and $\left\langle k\right\rangle$. Moreover, the observation that the
spectral radius scales with the average degree suggests that other spectral
magnitudes could also scale with $\left\langle k\right\rangle$, which is
precisely what is addressed in the next section.

##  III Energy of Erdős-Rényi digraphs

###  III.1 Short review of digraph energies

As mentioned in the Introduction, the definition of energy proposed by Gutman
(see Eq. (3)) cannot be directly applied to digraphs since, in this case, the
eigenvalues can be complex. However, by examining Eq. (3), a straightforward
generalization can be done by replacing the absolute value of the real
eigenvalues with the module of the complex eigenvalues of the adjacency matrix
of a digraph, here denoted as $Z_{k}$. In fact, this definition of energy,

$\mathcal{S}(D)=\sum_{k=1}^{n}|Z_{k}|,$ (12)

has been reported in Ref. MBG10 . Interestingly, this definition is not the
most widely studied. Instead of opting for this direct generalization, other
definitions have received more attention.

In Ref. PR08 , Peña and Rada, motivated by Coulson’s formula, generalized the
concept of energy of a digraph as

$e(D)=\sum_{k=1}^{n}\lvert\operatorname{Re}(Z_{k})\rvert.$ (13)

This digraph energy has been extensively studied; see for example Refs. R09 ;
B10 ; R10 ; R12 ; PBGR13 . Also, this definition has been extended to other
graph energies G12 ; LSG12 . Bounds have also been established for $e(D)$. For
example, in R09  Rada generalized McClelland’s inequality for directed graphs
with $n$ vertices, $m$ edges and $c_{2}$ closed walks of length $2$ as

$e(D)\leq\sqrt{\frac{1}{2}n\left(m+c_{2}\right)}.$ (14)

Also, a lower bound for $e(D)$ was established B10 ; R10 :

$e(D)\geq\sqrt{2c_{2}}.$ (15)

It is important to note that the definition of Peña and Rada, although it
satisfies the Coulson integral, does not consider the imaginary part of the
eigenvalues.

Another definition of energy of digraphs that does consider the imaginary part
of the eigenvalues was proposed by Khan, Farooq and Rada in Ref. KFR17 . This
is called the iota energy and is defined as

$E_{\iota}(D)=\sum_{k=1}^{n}\lvert\operatorname{Im}(Z_{k})\rvert.$ (16)

This energy can be defined from the Coulson integral formula using the
characteristic polynomial of the complex adjacency matrix $\mathbf{A}_{c}$.
Which is defined as

$\mathbf{A}_{c_{uv}}=\left\\{\begin{array}[]{ll}-\imath&\text{ if
}u\rightarrow v,\\\ 0&\text{ otherwise. }\end{array}\right.$ (17)

The iota energy has been extensively studied in digraphs with specific
characteristics such as bicyclic, tricyclic, and signed digraphs FKA17 ; YW18
; FKC19 ; FCK19 ; FK18 .

More recently, Khan proposed another energy definition that incorporates both
real and imaginary parts of the adjacency matrix eigenvalues. This energy is
called the p-energy and is defined as K21 ; KKA21

$E_{p}(D)=\sum_{k=1}^{n}\lvert\operatorname{Re}(Z_{k})\operatorname{Im}(Z_{k})\rvert.$
(18)

To represent this energy in an integral way with Coulson’s formula it is
necessary to use the characteristic polynomial of the squared adjacency matrix
$\mathbf{A}^{2}$ instead of the characteristic polynomial of $\mathbf{A}$.

Moreover, Nikiforov N07  proposed the concept of energy of a matrix using the
corresponding singular values. The singular values of a matrix are a set of
non-negative elements that are calculated from a matrix
$\mathbf{A}\in\mathcal{R}^{m\times n}$. They are defined as the square root of
the eigenvalues of the $\mathbf{A}^{T}\mathbf{A}\in\mathcal{R}^{n\times n}$
matrix. Given the singular values of a matrix $\mathbf{A}$, $\sigma_{k}$, the
energy of the matrix is defined as

$\mathcal{N}(\mathbf{A})=\sum_{k=1}^{n}\sigma_{k}.$ (19)

This concept has been widely studied for different types of matrices, such as
non-square matrices KT08  and for digraphs AR16 ; GMR21 . Its importance lies
in the fact that it can be computed for any matrix and, in the case of a
square symmetric matrix, it reproduces Eq. (3). For the Nikiforov energy, some
bounds have been reported in terms of the properties of the matrix.
Particularly, an upper bound for $\mathcal{N}(D)$ has been reported as B10 ;
KT08

$\mathcal{N}(D)\leq\frac{m}{n}+\sqrt{\left(n-1\right)\left(m-\frac{m^{2}}{n^{2}}\right)},$
(20)

while Agudelo and Rada proposed the lower bound AR16

$\mathcal{N}(D)\geq\sqrt{m}.$ (21)

Another definition of energy reported in the literature is the Hermitian
energy LL15 ; GM17 ; BP22 . In contrast to the previous definitions, this
energy is not directly linked to the adjacency matrix of the digraph. To
calculate this energy, it is necessary to construct the Hermitian adjacency
matrix, denoted as $\bf{H}$, which is defined as follows:

$H_{uv}=\left\\{\begin{array}[]{ll}1&\text{ if }u\leftrightarrow v,\\\
-\imath&\text{ if }u\rightarrow v,\\\ \imath&\text{ if }v\rightarrow u,\\\
0&\text{ otherwise. }\end{array}\right.$ (22)

Since $\bf{H}$ is a Hermitian matrix by construction, its eigenvalues are
real. Then, the Hermitian energy, denoted as $E_{H}(G)$, can be computed using
Eq. (3) with the eigenvalues of $\bf{H}$. Bounds have also been established
for $E_{H}(G)$. Considering $q$ as the determinant of $\bf{H}$ and $\Delta$
the maximum degree of the graph, the following bounds were derived LL15 :

$\sqrt{2m+n(n-1)q^{2/n}}\leq E_{H}(G)\leq n\sqrt{\Delta}.$ (23)

Additionally, when considering solely the number of arcs, an alternative bound
for the Hermitian energy is expressed as LL15

$2\sqrt{m}\leq E_{H}(G)\leq 2m.$ (24)

Particularly for ER digraphs, considering the relationships of $m$, $n$, and
$p$ found in the previous section, this bound can be expressed as

$2n\sqrt{p}\leq E_{H}(G)\leq 2n^{2}p.$ (25)

Although all these energy definitions have been extensively studied to
determine minimum and maximum bounds and have also been computed for specific
graphs, no numerical study has yet been performed to compare them. To fill
this gap, we have undertaken the task of numerically and statistically
evaluating these energies for ensembles of ER random digraphs.

###  III.2 Energies of ER digraphs

Here we compute the ${\cal S}(D)$ energy, the Peña-Rada energy $e(D)$, the
iota energy $E_{\iota}(D)$, the p-energy $E_{p}(D)$, the Nikiforov energy
$\mathcal{N}(D)$, and the Hermitian energy $E_{H}(D)$ for ensembles of ER
digraphs characterized by the parameter pair $(n,p)$.

Then, in Fig. 7 we plot the average energies of ER digraphs of size $n=100$ as
a function of the connection probability $p$. In Fig. 7 we also indicate the
bounds given by Eqs. (14), (20), (21) and (25), and the hyperenergetic limit
$2n-2$.

Figure 7: Average energies as a function of the connection probability $p$ of
Erdős-Rényi digraphs of size $n=100$. The dashed lines indicate the limits
defined by equations (14) (brown color), (20) (maroon color), (21) (green
color) and the upper and lower bounds given by Eq. (25) (violet and cyan,
respectively). The black dotted line indicates the hyperenergetic limit. The
vertical red-dotted line is $p=\frac{1}{2}(1+\frac{1}{\sqrt{n}})$ with
$n=100$. Each symbol was calculated by averaging over $10^{6}/n$ random
digraphs.

From Fig. 7, we can see that all energies exhibit a similar pattern as a
function of $p$: As $p$ increases, the energy increases until reaching a
maximum value at $p$ close to 1; then, it decreases displaying a bell-like
shape that is better observed in a semi-logarithmic scale. However, the
maximum values for different energies are different. We numerically computed
the maximum values reached by the different energies for ensembles of digraphs
of different sizes (not shown here). We found that all energies reach their
maximum at $p\approx 0.5$. Moreover, we can see in Fig. 7 that the curve
corresponding to Eq. (20) also reaches its maximum at $p\approx 0.5$, in
agrrement with all the numerically computed energies. Then, in order to get an
estimation of the value of $p$ producing the energy maxima we rewrite Eq. (20)
in terms of $n$ and $p$ using $m\approx n^{2}p$, then we see that

$\displaystyle\mathcal{N}(D)\leq np+\sqrt{(n-1)(n^{2}p-n^{2}p^{2})}$
$\displaystyle=\ np+n\sqrt{(n-1)p(1-p)}.$ (26)

So we find that the maximum of Eq. (26) occurs at
$p=\frac{1}{2}(1+\frac{1}{\sqrt{n}})$, which is consistent with the numerical
observation.

Figure 8: Average energy $\left\langle{\cal S}(D)\right\rangle$, (b) average
Peña-Rada energy $\left\langle e(D)\right\rangle$, (c) average iota energy
$\left\langle E_{\iota}(D)\right\rangle$, (d) average p-energy $\left\langle
E_{p}(D)\right\rangle$, (e) average Nikiforov energy
$\left\langle\mathcal{N}(D)\right\rangle$ and (f) average hermitian energy
$\left\langle E_{H}(D)\right\rangle$ as a function of the connection
probability $p$ of Erdős-Rényi digraphs of sizes $n\in[50,400]$. Each symbol
was calculated by averaging over $10^{6}/n$ random digraphs.

We recall that in Fig. 7 we used ER digraphs of size $n=100$ so, to see the
effect of the graph size on the average energies, in Fig. 8 we plot them as a
function of $p$ for different values of $n$. From this figure we can observe
that the curves for a given energy definition exhibit a similar functional
dependence of $p$ but they are shifted on both axis for increasing $n$. This
effect of $n$ on the energies is equivalent to that observed in the previous
Section for the topological and spectral properties of ER digraphs, see Figs.
1 and 5. Thus, taking as a reference the scaling analysis of Sec. II, in Fig.
9 we plot the energies normalized to $n$ now as a function of the average
degree. Indeed, we observe that $\left\langle k\right\rangle$ works well as
the scaling parameter of the normalized energies, mainly above the percolation
threshold $\left\langle k\right\rangle>1$. Moreover, remarkably, $\left\langle
k\right\rangle$ perfectly scales the normalized Nikiforov energy as well as
the normalized Hermitian energy over the entire range of connection
probabilities, see Figs. 9(e,f).

In addition, in Fig. 9 we can see that certain energy pairs depend on
$\left\langle k\right\rangle$ in a very similar way and, consequently, they
should be strongly correlated. Specifically, we observe strong similarities
between ${\cal S}(D)$ and $e(D)$, see Figs. 9(a,b); and between
$\mathcal{N}(D)$ and $E_{H}(D)$, see Figs. 9(e,f). So, in Fig. 10 we present
scatter plots of these pairs of energies and report the corresponding Pearson
correlation coefficients. Moreover, the strong correlations reported in Fig.
10 allowed us to state the following relations:

$\displaystyle\sqrt{2}\ e(D)$ $\displaystyle\approx$ $\displaystyle{\cal
S}(D),$ (27) $\displaystyle E_{H}(D)$ $\displaystyle\approx$
$\displaystyle\frac{8}{5}\ \mathcal{N}(D),$ (28)

see the black-dashed lines Fig. 10.

Figure 9: (a) $\left\langle{\cal S}(D)\right\rangle$, (b) $\left\langle
e(D)\right\rangle$, (c) $\left\langle E_{\iota}(D)\right\rangle$, (d)
$\left\langle E_{p}(D)\right\rangle$, (e)
$\left\langle\mathcal{N}(D)\right\rangle$ and (f) $\left\langle
E_{H}(D)\right\rangle$ normalized to $n$ as a function of the average degree
$\left\langle k\right\rangle$ of Erdős-Rényi digraphs of sizes $n\in[50,400]$.
Same data sets of Fig. 8. Figure 10: Scatter plots of (a) $\left\langle{\cal
S}(D)\right\rangle$ vs. $\left\langle e(D)\right\rangle$ and (b)
$\left\langle\mathcal{N}(D)\right\rangle$ vs. $\left\langle
E_{H}(D)\right\rangle$. Data corresponds to $n=50$ and 400. The Pearson
correlation coefficients $r$ are reported in the corresponding panels. The
black-dashed lines are fittings of the form $y={\cal C}x$ with (a) ${\cal
C}=1/\sqrt{2}$, and (b) ${\cal C}=\frac{8}{5}$.

##  IV Conclusions and discussion

This study aims to contribute to the understanding of topological and spectral
properties of random digraphs. Specifically, we studied some topological and
spectral properties of Erdős-Rényi (ER) digraphs $D(n,p)$.

Initially, we focused on the statistical analysis of topological properties by
computing the average number of non-isolated vertices $\left\langle
V_{x}(D)\right\rangle$, the average Randić index $\left\langle
R(D)\right\rangle$ and the average sum-connectivity index
$\left\langle\chi(D)\right\rangle$. By means of a scaling analysis, we found
the total average degree $\left\langle k\right\rangle$ works well as scaling
parameter of $\left\langle V_{x}(D)\right\rangle$, $\left\langle
R(D)\right\rangle$ and $\left\langle\chi(D)\right\rangle$ but also for the
average number of arcs $\left\langle m(D)\right\rangle$, the average spectral
radius $\left\langle\rho(D)\right\rangle$ and the average closed walks of
length 2 $\left\langle c_{2}(D)\right\rangle$. Moreover, we were able to infer
the following relations: $\left\langle m(D)\right\rangle/n\approx\left\langle
k\right\rangle/2$, $\left\langle c_{2}(D)\right\rangle\approx\left\langle
k\right\rangle^{2}/8$, and
$\left\langle\rho(D)\right\rangle\approx\left\langle k\right\rangle/2$ for
$\left\langle k\right\rangle>1$.

Concerning spectral properties, we computed six different graph energies for
ensembles of ER digraphs $D(n,p)$: the ${\cal S}(D)$ energy, the Peña-Rada
energy $e(D)$, the iota energy $E_{\iota}(D)$, the p-energy $E_{p}(D)$, the
Nikiforov energy $\mathcal{N}(D)$, and the Hermitian energy $E_{H}(D)$. First,
we showed that $\left\langle k\right\rangle$ scales well all normalized
averaged energies, mainly above the percolation threshold $\left\langle
k\right\rangle>1$. Moreover, remarkably, $\left\langle k\right\rangle$
perfectly scales $\left\langle\mathcal{N}(D)\right\rangle/n$ and $\left\langle
E_{H}(D)\right\rangle/n$ over the entire range of connection probabilities.
Then, we reformulated a set of bounds previously reported in the literature
for these energies as a function $(n,p)$. So, by identifying strong
correlations between ${\cal S}(D)$ and $e(D)$, and between $\mathcal{N}(D)$
and $E_{H}(D)$ we phenomenologically stated linear relations between energies,
see Eqs. (27-28).

It is important to stress that Eqs. (27-28) can be used to extend previously
known bounds. That is, from Eqs. (14-15) and (27) we get

$2\sqrt{c_{2}}\leq S(D)\leq\sqrt{n\left(m+c_{2}\right)},$

by combining Eqs. (20-21) and (28) we can write

$\frac{8}{5}\sqrt{m}\leq
E_{H}(D)\leq\frac{8}{5}\left[\frac{m}{n}+\sqrt{\left(n-1\right)\left(m-\frac{m^{2}}{n^{2}}\right)}\right]$

while from Eq. (24) and (28) we obtain

$\frac{5}{4}\sqrt{m}\leq\mathcal{N}(D)\leq\frac{5}{4}m.$

Which in the particular case of ER digraphs, read as:

$\sqrt{2}np\leq S(D)\leq n\sqrt{np\left(1+\frac{p}{2}\right)},$ (29)
$\frac{8}{5}n\sqrt{p}\leq
E_{H}(D)\leq\frac{8}{5}n\left[p+\sqrt{\left(n-1\right)p\left(1-p\right)}\right]$
(30)

and

$\frac{5}{4}n\sqrt{p}\leq\mathcal{N}(D)\leq\frac{5}{4}n^{2}p,$ (31)

respectively. Finally, in Fig. 11, we validate Eqs. (29-31) on ER digraphs of
size $n=400$. We just note that the lower bound in Eq. (29) fails to bound
$\left\langle{\cal S}(D)\right\rangle$, see Fig. 11(a); however this also
happens in the original Eq. (15).

Figure 11: (a) $\left\langle{\cal S}(D)\right\rangle$, (b)
$\left\langle\mathcal{N}(D)\right\rangle$ and (c) $\left\langle
E_{H}(D)\right\rangle$ as a function of the connection probability $p$ of
Erdős-Rényi digraphs of size $n=400$. The dashed lines in panels (a), (b) and
(c) indicate the limits given by Eqs. (29), (30) and (31), respectively.

###### Acknowledgements.

J.A.M.-B. thanks support from CONACyT (Grant No. 286633), CONACyT-Fronteras
(Grant No. 425854), and VIEP-BUAP (Grant No. 100405811-VIEP2023), Mexico. C.
T. M. M. thanks support from CONAHCYT (CVU No. 784756).

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