Related papers: The spectral radius of graphs without paths and cy…

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…

Let G be a graph of n vertices and m edges, and let G has no cycles of length 4. We give upper bounds on the adjacency spectral radius of G in terms of n and m.

Let $G$ be a graph with $n$ vertices and $\lambda_n(G)$ be the least eigenvalue of its adjacency matrix of $G$. In this paper, we give sharp bounds on the least eigenvalue of graphs without given pathes or cycles and determine the extremal…

Let G be a graph of given order and mu(G) be the largest eigenvalue of its adjacency matrix. We give conditions on mu(G) that imply Hamiltonicity of G and of its complement.

It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we…

Let $H_{s,t_1,\ldots ,t_k}$ be the graph with $s$ triangles and $k$ odd cycles of lengths $t_1,\ldots ,t_k\ge 5$ intersecting in exactly one common vertex. Recently, Hou, Qiu and Liu [Discrete Math. 341 (2018) 126--137], and Yuan [J. Graph…

The signless Laplacian spectral radius of a graph $G$, denoted by $q(G)$, is the largest eigenvalue of its signless Laplacian matrix. In this paper, we investigate extremal signless Laplacian spectral radius for graphs without short cycles…

In this paper, we study the maximum adjacency spectral radii of graphs of large order that do not contain an even cycle of given length. For $n>k$, let $S_{n,k}$ be the join of a clique on $k$ vertices with an independent set of $n-k$…

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 an irregular graph on $n$ vertices with maximum degree $\Delta$ and diameter $D$. We show that \Delta-\lambda_1>\frac{1}{nD} where $\lambda_1$ is the largest eigenvalue of the adjacency matrix of $G$. We also study the effect of…

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…

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) +…

Let $G$ be a graph with minimum degree $\delta$. The spectral radius of $G$, denoted by $\rho(G)$, is the largest eigenvalue of the adjacency matrix of $G$. In this note we mainly prove the following two results. (1) Let $G$ be a graph on…

Let $t\geq3$ and $G$ be a graph of order $n,$ with no $K_{2,t}$ minor. If $n>400t^{6}$, then the spectral radius $\mu\left( G\right) $ satisfies \[ \mu\left( G\right) \leq\frac{t-1}{2}+\sqrt{n+\frac{t^{2}-2t-3}{4}}, \] with equality if and…

Let G be a simple connected graph of order n with degree sequence d_1, d_2, ..., d_n in non-increasing order. The spectral radius rho(G) of G is the largest eigenvalue of its adjacency matrix. For each positive integer L at most n, we give…

The spectral radius of a graph is the largest modulus of an eigenvalue of its adjacency matrix. Let $\mathcal{C}_{n, e}$ be the set of all the connected simple graphs with $n$ vertices and $n - 1 + e$ edges. Here, we solve the spectral…

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 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…

For a graph $G$, the spectral radius $\lambda_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all…

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.…