Related papers: On the Maximum ABC Spectral Radius of Connected Gr…

Let $G$ be a graph of size $m$ and $\rho(G)$ be the spectral radius of its adjacency matrix. A graph is said to be $F$-free if it does not contain a subgraph isomorphic to $F$. In this paper, we prove that if $G$ is a $K_{2,r+1}$-free…

The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the $|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if…

For a connected graph $G$, let $\mu(G)$ denote the distance spectral radius of $G$. A matching in a graph $G$ is a set of disjoint edges of $G$. The maximum size of a matching in $G$ is called the matching number of $G$, denoted by…

Let $m$ be a positive integer. Brualdi and Hoffman proposed the problem to determine the (connected) graphs with maximum spectral radius in a given graph class and they posed a conjecture for the class of graphs with given size $m$. After…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For any real $\alpha\in [0,1]$, Nikiforov [Merging the $A$- and $Q$-spectral theories, Appl. Anal. Discrete Math. 11 (2017)…

For a finite, simple, and undirected graph $G$ with $n$ vertices, $m$ edges, and largest eigenvalue $\lambda$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-\frac{2m}{n}\right|$. Contributing to a…

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=14$ for $\alpha\in[0,\frac{1}{2}]$, $f(\alpha)=17$ for $\alpha\in(\frac{1}{2},\frac{2}{3}]$, $f(\alpha)=20$ for…

Let $\lambda_1,\dots,\lambda_n$ be the eigenvalues of a graph $G$. For any $k\geq 0$, the $k$-th spectral moment of $G$ is defined by $\M_k(G)=\lambda_1^k+\dots+\lambda_n^k$. We use the fact that $\M_k(G)$ is also the number of closed walks…

We show that for a graph $G$ with the vertex set $V$ and the largest eigenvalue $\lambda_{\max}(G)$, letting $$ M(G) := \max_{X,Y \subset V} \frac{e(X,Y)}{\sqrt{|X||Y|}} $$ (where $e(X,Y)$ denotes the number of edges between $X$ and $Y$),…

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the…

Let $G$ be a graph. The spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. For an integer $k\geq1$, a $k$-factor of $G$ is a $k$-regular spanning subgraph of $G$. Assume that $k$ and $n$ are integers…

The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral…

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The…

Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of vertex degrees of $G$. For any real $\alpha \in [0,1]$, Nikiforov defined the $A_\alpha$-matrix of a graph $G$ as $A_\alpha(G)=\alpha…

Let $A(G)$ and $D(G)$ be the adjacency and degree matrices of a simple graph $G$ on $n$ vertices, respectively. The \emph{$A_\alpha$-spectral radius} of $G$ is the largest eigenvalue of $A_\alpha (G)=\alpha D(G)+(1-\alpha)A(G)$ for a real…

The spanning tree packing number of a graph $G$, denoted by $\tau(G)$, is the maximum number of edge-disjoint spanning trees contained in $G$. The study of $\tau(G)$ is one of the classic problems in graph theory. Cioab\u{a} and Wong…

Let G be a graph with n vertices and mu(G) be the largest eigenvalue of the adjacency matrix of G. We study how large mu(G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral…

For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of…

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…