Related papers: A note on the $A_{\alpha}$-spectral radius of grap…

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the…

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…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right],$ define the matrix $A_{\alpha}\left(G\right) $ as \[ A_{\alpha}\left(G\right) =\alpha…

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1].…

Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} introduced the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ for…

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue…

For a given graph \( G \), let \( A(G) \), \( Q(G) \), and \( D(G) \) denote the adjacency matrix, signless Laplacian matrix, and diagonal degree matrix of \( G \), respectively. The \( A_\alpha(G) \) matrix, proposed by Nikiforov, is…

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_{\alpha}$-matrix: $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where…

For a graph $G$ and for two distinct vertices $u$ and $v$, let $\kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written…

Let G be a graph with adjacency matrix A(G) and degree diagonal matrix D(G). In 2017, Nikiforov [1] defined the matrix Aalpha(G), as a convex combination of A(G) and D(G), the following way, Aalpha(G) = alpha A(G) + (1 - alpha)D(G), where…

Let $\alpha\in[0,1)$, and let $G$ be a graph of even order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=10$ for $0\leq \alpha\leq1/2$, $f(\alpha)=14$ for $1/2<\alpha\leq 2/3$ and $f(\alpha)=5/(1-\alpha)$ for $2/3<\alpha<1$. In this paper,…

The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The…

For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{v_{i}}$ (or $d_{i}$ for short) the degree of vertex $v_{i}$. The $p$-Sombor matrix $\textbf{S}_{\textbf{p}}(G)$ ($p\neq0$) of a graph $G$ is a square matrix, where the $(i,j)$-entry is…

Nikiforov conjectured that for a given integer $k\ge 2$, any graph $G$ of sufficiently large order $n$ with spectral radius $\mu(G)\geq \mu(S_{n,k})$ (or $\mu(G)\ge \mu(S_{n,k}^+))$ contains $C_{2k+1}$ or $C_{2k+2}$(or $C_{2k+2}$), unless…

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 collection of…

For $0 \leq \alpha < 1$, the $\alpha$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$,…

The $\alpha$-spectral radius of a connected graph $G$ is the spectral radius of $A_\alpha$-matrix of $G$. In this paper, we discuss the methods for comparing $\alpha$-spectral radius of graphs. As applications, we characterize the graphs…

Let $\mu(G)$ denote the spectral radius of a graph $G$. We partly confirm a conjecture due to Nikiforov, which is a spectral radius analogue of the well-known Erd\H{o}s-S\'os Conjecture that any tree of order $t$ is contained in a graph of…

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…

For real $\alpha\in [0,1)$ and a hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric…