Related papers: On the $A_{\alpha}$ and $RD_{\alpha}$ matrices ove…

Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix of a digraph $G$ is defined as $$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G),$$ where $A(G)$ is the adjacency matrix of…

Within a random-matrix-theory approach, we use the nearest-neighbor energy level spacing distribution $P(s)$ and the entropic eigenfunction localization length $\ell$ to study spectral and eigenfunction properties (of adjacency matrices) of…

Let $G$ be a digraph and $A(G)$ be the adjacency matrix of $G$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. For any real $\alpha\in[0,1]$, Liu et al. \cite{LWCL} defined the matrix $A_\alpha(G)$ as…

The power graph $\mathcal{P}(G)$ of a given finite group $G$ is the simple undirected graph whose vertices are the elements of $G$, in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power…

The power graph of a group $G$ is the graph whose vertex set is $G$ and two distinct vertices are adjacent if one is a power of the other. In this paper, the minimum degree of power graphs of certain classes of cyclic groups, abelian…

The Difference graph $\mathcal{D}(G)$ of a finite group $G$ is the difference of the enhanced power graph $\mathcal{P}_{E}(G)$ and the power graph $\mathcal{P}(G)$ with all the isolated vertices removed. In this paper, we characterize the…

Let $G$ be a connected graph of order $n$, and $A(G)$ and $D(G)$ its adjacency and degree diagonal matrices, respectively. For a parameter $\alpha \in [0,1]$, Nikiforov~(2017) introduced the convex combination $A_{\alpha}(G) = \alpha D(G) +…

Let $A_\alpha(G)$ be the $A_\alpha$-matrix of a digraph $G$ and $\lambda_{\alpha 1}, \lambda_{\alpha 2}, \ldots, \lambda_{\alpha n}$ be the eigenvalues of $A_\alpha(G)$. Let $\rho_\alpha(G)$ be the $A_\alpha$ spectral radius of $G$ and…

The \emph{difference subgroup graph} $D(G)$ of a finite group $G$ is defined as the graph whose vertices are the non-trivial proper subgroups of $G$, with two distinct vertices $H$ and $K$ adjacent if and only if $\langle H, K \rangle = G$…

The \emph{power graph} $P(G)$ of a group $G$ is the graph whose vertex set is $G$, with $x$ and $y$ joined if one is a power of the other; the \emph{directed power graph} $\vec{P}(G)$ has the same vertex set, with an arc from $x$ to $y$ if…

For a simple graph $G$, the generalized adjacency matrix $A_{\alpha}(G)$ is defined as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G), \alpha\in [0,1]$, where $A(G)$ is the adjacency matrix and $D(G)$ is the diagonal matrix of vertex degrees of…

In this article, we analyze the limiting eigenvalue distribution (LED) of random geometric graphs (RGGs). The RGG is constructed by uniformly distributing $n$ nodes on the $d$-dimensional torus $\mathbb{T}^d \equiv [0, 1]^d$ and connecting…

Let $G$ be a graph with $n$ vertices, and let $A(G)$ and $D(G)$ denote respectively the adjacency matrix and the degree matrix of $G$. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The…

For a graph $\Gamma$ and group $G$, $G^\Gamma$ is the subgroup of $G^{|\Gamma|}$ generated by elements with $g$ in the coordinates corresponding to $v$ and its neighbors in $\Gamma$. There is a natural epimorphism $G^\Gamma \to…

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G)…

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph $G$ has exactly two main eigenvalues if and only if there exists a unique pair of…

The power graph of a given finite group is a simple undirected graph whose vertex set is the group itself, and there is an edge between any two distinct vertices if one is a power of the other. In this paper, we find a precise formula to…

Given a finite group $G$, the character graph, denoted by $\Delta(G)$, for its irreducible character degrees is a graph with vertex set $\rho(G)$ which is the set of prime numbers that divide the irreducible character degrees of $G$, and…

The power graph and the enhanced power graph of a group $\mathbf G$ are simple graphs with vertex set $G$; two elements of $G$ are adjacent in the power graph if one of them is a power of the other, and they are adjacent in the enhanced…

For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $\Gamma_H(G)$ whose vertex set $V(\Gamma_H(G))=(G\setminus H)\bigcup \{e\}$ and two vertices $a$ and $b$ are edge connected if…