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A wheel graph is a graph formed by connecting a single vertex to all vertices of a cycle. A graph is called wheel-free if it does not contain any wheel graph as a subgraph. In 2010, Nikiforov proposed a Brualdi-Solheid-Tur\'{a}n type…

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…

Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most $k$, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order $n$ with $\kappa(G) \leq k$ and showed that among those graphs, the…

The normalized distance Laplacian of a graph $G$ is defined as $\mathcal{D}^\mathcal{L}(G)=T(G)^{-1/2}(T(G)-\mathcal{D}(G))T(G)^{-1/2}$ where $\mathcal{D}(G)$ is the matrix with pairwise distances between vertices and $T(G)$ is the diagonal…

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.

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)…

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 spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the…

A graph $G$ is said to be $F$-free if it does not contain $F$ as a subgraph. A theta graph, say $\theta_{l_1,l_2,l_3}$, is the graph obtained by connecting two distinct vertices with three internally disjoint paths of length $l_1, l_2,…

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…

The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear…

For a $hypergraph$ $\mathcal{G}=(V, E)$ with 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 $(\mathcal…

In this paper we show how the $\alpha$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $\alpha$-spectral radius among $k$-uniform non-caterpillar…

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…

This paper presents a sharp upper bound for the spectral radius of simple digraphs with described number of arcs. Further, the extremal graphs which attain the maximum spectral radius among all simple digraphs with fixed arcs are…

In this paper, we determine the graphs whose spectral radius and distance spectral radius attain maximum and minimum among all complements of clique trees. Furthermore, we also determine the graphs whose spectral radius and distance…

In this paper, we determine the maximal Laplacian and signless Laplacian spectral radii for graphs with fixed number of vertices and domination number, and characterize the extremal graphs respectively.

Let $\mathcal{U}(n,k)$ and $\Gamma(n,k)$ be the set of the $k$-uniform linear and nonlinear unicyclic hypergraphs having perfect matchings with $n$ vertices respectively, where $n\geq k(k-1)$ and $k\geq 3$. By using some techniques of…

Let $G$ be a graph. We say that a hypergraph $H$ is a Berge-$G$ if there is a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq \phi(e)$ for all $e\in E(G)$. For any $r$-uniform hypergraph $H$ and a real number $p\geq 1$, the…

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…