Related papers: The existence of the graphs that have exactly two …

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 $\mu_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $\kappa(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $\mu_2(G) \le…

Given a simple graph $G$, its $A_\alpha$ matrix is a convex combination with parameter $\alpha\in [0,1]$ of its adjacency matrix and its degree diagonal matrices. Here we compare two lower bounds presented in [J. D. G. Silva Jr., C. S.…

The double graph of a graph $G$ is defined as $\mathcal{D}[G]$ = $G \times T_2$, where \(T_2\) is the total graph with 2 vertices and $\times$ stands for the Kronecker product of graphs. In this paper, sufficient conditions for double…

The investigation of eigenvalue conditions for the existence of an $[a,b]$-factor originates in the work of Brouwer and Haemers (2005) on perfect matchings. In the decades since, spectral extremal problems related to $[a,b]$-factors have…

The Laplacian matrix of a graph $G$ is denoted by $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two graphs. A one-edge connection of…

Energy of a simple graph $G$, denoted by $\mathcal{E}(G)$, is the sum of the absolute values of the eigenvalues of $G$. Two graphs with the same order and energy are called equienergetic graphs. A graph $G$ with the property $G\cong…

A vertex with neighbours of degrees $d_1 \geq ... \geq d_r$ has {\em vertex type} $(d_1, ..., d_r)$. A graph is {\em vertex-oblique} if each vertex has a distinct vertex-type. While no graph can have distinct degrees, Schreyer, Walther and…

Let $G=(V(G),E(G)) $ be a graph with vertex set $V(G)$ and edge set $E(G)$. An even factor of $G$ is a spanning subgraph $F$ such that every vertex in $F$ has a nonzero even degree. Note that $\delta(G)\geq 2$ is a trivial necessary…

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. The vicinal preorder on the vertex set of a graph is defined in terms of inclusions among the neighborhoods of vertices. The minimum number of chains…

A subset $S\subseteq V$ in a graph $G=(V,E)$ is a total $[1,2]$-set if, for every vertex $v\in V$, $1\leq |N(v)\cap S|\leq 2$. The minimum cardinality of a total $[1,2]$-set of $G$ is called the total $[1,2]$-domination number, denoted by…

A vertex subset $S$ of a graph $G$ is a dominating set if every vertex of $G$ either belongs to $S$ or is adjacent to a vertex of $S$. The cardinality of a smallest dominating set is called the dominating number of $G$ and is denoted by…

In this work, we introduce the boundary polynomial of a graph $G$ as the ordinary generating function in two variables $B(G;x,y):= \displaystyle\sum_{S\subseteq V(G)} x^{|B(S)|}y^{|S|}$, where $B(S)$ denotes the outer boundary of $S$. We…

A non-empty set $S\subseteq V (G)$ of the simple graph $G=(V(G),E(G))$ is an independent dominating set of $G$ if every vertex not in $S$ is adjacent with some vertex in $S$ and the vertices of $S$ are pairwise non-adjacent. The independent…

A set $S\subseteq V$ is \textit{independent} in a graph $G=\left( V,E\right) $ if no two vertices from $S$ are adjacent. The \textit{independence number} $\alpha(G)$ is the cardinality of a maximum independent set, while $\mu(G)$ is the…

In this note we show that the minimum number of distinct eigenvalues of a threshold graph is at most $4$. Moreover, given any threshold graph $G$ and any nonzero real number $\lambda$, we explicitly construct a matrix $M$ associated with…

A nut graph is a simple graph whose adjacency matrix has the eigenvalue~0 with multiplicity~1 such that its corresponding eigenvector has no zero entries. Motivated by a question of Fowler et al.~[\emph{Disc. Math. Graph Theory} 40 (2020),…

An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $d_H(v)$ is odd and $1\le d_H(v) \le b$. Let $\lambda_3(G)$ be the third largest eigenvalue of the adjacency matrix of $G$. For positive…

Let $\G(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1, >..., d_v$, of the vertices of $G$. It is known that the maximum value of $P_2(G)$…

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 \cite{VN1} defined the matrix $A_{\alpha}(G)$ as $$A_{\alpha}(G)=\alpha…