Related papers: Extremal $ABS$ Spectral Radius in Bicyclic and Bip…

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…

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) +…

For a $hypergraph$ $\mathcal{G}=(V, E)$ consisting of a nonempty vertex set $V=V(\mathcal{G})$ and an edge set $E=E(\mathcal{G})$, its $adjacency$ $matrix$ $\mathcal {A}_{\mathcal{G}}=[(\mathcal {A}_{\mathcal{G}})_{ij}]$ is defined as…

Bicyclic graph is a connected graph in which the number of edges equals the number of vertices plus one. In this paper, we determine the graph which alone maximizes the spectral radii among all the bicyclic graphs on $n$ vertices with fixed…

Here we study the spectral radii of some linear hypergraphs, that is, the maximum moduli of the eigenvalues of their corresponding adjacency matrices. We determine the hypertrees having the largest to seventh-largest spectral radii. The…

For a graph $G$, the spectral radius $\rho(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. In this paper, we give three lammas on $\rho(G)$ when $G$ contains a spanning complete bipartite graph. Using these lemmas and typical…

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…

Let $d_{v_{i}}$ be the degree of the vertex $v_{i}$ of $G$. The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle…

A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. Let $\mathcal{B}(\mathbf{k}, \alpha)$ be the class of bi-block graph on $\mathbf{k}$ vertices with given independence number $\alpha$. It is…

Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $\mu(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without…

In this paper we investigate the hypergraphs whose spectral radii attain the maximum among all uniform hypergraphs with given number of edges. In particular we characterize the hypergraph(s) with maximum spectral radius over all unicyclic…

A bipartite graph is subcubic if it is an irregular bipartite graph with maximum degree three. In this paper, we prove that the asymptotic value of maximum spectral radius over subcubic bipartite graphs of order $n$ is…

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$ and $m=|E|$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $\Delta=\max_{1\leq i \leq n} d_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times…

Let $G$ denote a bipartite graph with $e$ edges without isolated vertices. It was known that the spectral radius of $G$ is at most the square root of $e$, and the upper bound is attained if and only if $G$ is a complete bipartite graph.…

Spectral radius of a graph $G$ is the largest eigenvalue of adjacency matrix of $G$. The least eigenvalue of a graph $G$ is the least eigenvalue of adjacency matrix of $G$. In this paper we determine the graphs which attain respectively the…

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…

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be a diagonal matrix of the degrees of $G$. In 2017, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as \begin{equation*} A_{\alpha}(G)=\alpha G)+(1-\alpha)A(G),…

We give a bound on the spectral radius of subgraphs of regular graphs with given order and diameter. We give a lower bound on the smallest eigenvalue of a nonbipartite regular graph of given order and diameter.

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

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].…