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Let $\alpha(G)$ and $\beta(G)$, denote the size of a largest independent set and the clique cover number of an undirected graph $G$. Let $H$ be an interval graph with $V(G)=V(H)$ and $E(G)\subseteq E(H)$, and let $\phi(G,H)$ denote the…

Let $G$ be a connected graph with $n$ vertices. The isolated toughness of $G$, denoted by $I(G)$, is defined by $I(G)=\min\left\{\frac{|S|}{i(G-S)}:S\subseteq V(G) \ \mbox{and} \ i(G-S)\geq2\right\}$ if $G$ is not complete, or…

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…

For $0\leq \alpha < 1$, the $\mathcal{A}_{\alpha}$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_{\alpha}(G):=\alpha \mathcal{D}(G)+(1-\alpha) \mathcal{A}(G)$, where…

For any $\alpha\in (0,1)$ and any $n^{\alpha}\leq d\leq n/2$, we show that $\lambda(G)\leq C_\alpha \sqrt{d}$ with probability at least $1-\frac{1}{n}$, where $G$ is the uniform random $d$-regular graph on $n$ vertices, $\lambda(G)$ denotes…

For a finite, simple, and undirected graph $G$ with $n$ vertices and average degree $d$, Nikiforov introduced the degree deviation of $G$ as $s=\sum_{u\in V(G)}\left|d_G(u)-d\right|$. Provided that $G$ has largest eigenvalue $\lambda$,…

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…

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…

The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{i}+d_{j}}{2\sqrt{d_{i}d_{j}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The…

For an $n \times n$ matrix $A$, let $q(A)$ be the number of distinct eigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let $\mathcal{S}(G)$ be the set of all real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for…

We prove that every $n$-vertex directed graph $G$ with the minimum outdegree $\delta^+(G) = d$ contains a subgraph $H$ satisfying \[ \min\left\{\delta^+(H), \delta^-(H) \right\} \ge \frac{d(d+1)}{2n} \,.\] We also show that if $d = o(n)$…

Let G be a finite group acting orthogonally on a pair (S^d,\Gamma) where \Gamma is a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y.…

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 $G$ be a graph without isolated vertices and let $\alpha(G)$ be its stability number and $\tau(G)$ its covering number. The {\it $\alpha_{v}$-cover} number of a graph, denoted by $\alpha_{v}(G)$, is the maximum natural number $m$ such…

Let $G$ be a connected graph of order $n$. A $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$-factor of $G$ is a spanning subgraph of $G$ such that each component is isomorphic to a member in $\{P_{2},C_{3},P_{5},\mathcal{T}(3)\}$, where…

Let $\lambda_2(G)$ and $\kappa'(G)$ be the second largest eigenvalue and the edge-connectivity of a graph $G$, respectively. Let $d$ be a positive integer at least 3. For $t=1$ or 2, Cioaba proved sharp upper bounds for $\lambda_2(G)$ in a…

Let $A(G)$ be the adjacent matrix and $D(G)$ the diagonal matrix of the degrees of a graph $G$, respectively. For $0 \leq \alpha \leq 1$, the $A_{\alpha}$ matrix $A_{\alpha}(G) = \alpha D(G) +(1-\alpha)A(G)$ is given by Nikiforov. Clearly,…

The power graph $G = P(\Omega)$ of a finite group $\Omega$ is a graph with the vertex set $\Omega$ and two vertices $u, v \in \Omega$ form an edge if and only if one is an integral power of the other. Let $D(G)$, $A(G)$, $RT(G)$, and…

In a graph $G$, a vertex dominates itself and its neighbors. A subset $S$ of vertices of $G$ is a double dominating set of $G$ if every vertex is dominated by at least two vertices in $S$. The double domination number $\gamma_{\times 2}(G)$…

The dissociation number ${\rm diss}(G)$ of a graph $G$ is the maximum order of a set of vertices of $G$ inducing a subgraph that is of maximum degree at most $1$. Computing the dissociation number of a given graph is algorithmically hard…