Related papers: Maximum Estrada Index of Bicyclic Graphs

Let $G = (V, E)$ be a graph. We define matrices $M(G; \alpha, \beta)$as $\alpha D + \beta A$, where $\alpha$, $\beta$ are real numbers such that $(\alpha, \beta) \neq (0, 0)$ and $D$ and $A$ are the diagonal matrix and adjacency matrix of…

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and…

The sum $\lambda_1 + \lambda_n$ of the maximum and minimum eigenvalues, and the odd girth of a graph both measure bipartiteness. We seek to relate these measures. In particular, for an odd integer $k\geq 3$, let $\gamma_k$ denote the…

Let $r(u,v)$ be the resistance distance between two vertices $u, v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$…

For a graph $G$, let $\lambda_1(G)$ and $\lambda_2(G)$ denote the largest and the second largest adjacency eigenvalue of $G$. The sum $\lambda_1(G) + \lambda_2(G)$ is called the \emph{spectral sum} of $G$. We investigate the spectral sum of…

F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest…

In this paper we introduce a parameter $Mm(G)$, defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph $G$. We compute $Mm(G)$ for several families of graphs.

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $\cal D$ of…

Woodall proved that for a graph $G$ of order $n\geq 2k+3$ where $k\geq 0$ is an integer, if $e(G)\geq \binom{n-k-1}{2}+\binom{k+2}{2}+1$ then $G$ contains a $C_{\ell}$ for each $\ell\in [3,n-k]$. In this article, we prove a stability result…

Let $G$ be a graph and let $I := I (G)$ be its edge ideal. In this paper, we provide an upper bound of $n$ from which $\depth R/ I(G)^n$ is stationary, and compute this limit explicitly. This bound is always achieved if $G$ has no cycles of…

For a simple graph $G$, the energy $\mathcal{E}(G)$ is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix $A(G)$. Let $n, m$, respectively, be the number of vertices and edges of $G$. One well-known…

Let $\lambda^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $\Delta$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow…

Let $G=(V, E)$ be a simple graph with vertex set $V$ and edge set $E$. The Sombor index of the graph $G$ is a degree-based topological index, defined as $$SO(G)=\sum_{uv \in E}\sqrt{d(u)^2+d(v)^2},$$ in which $d(x)$ is the degree of the…

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…

For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In…

The general Sombor index of $G$ is defined as $SO_{\alpha}(G)= \sum_{uv\in G}\left(d^2_{G}(u)+d^2_{G}(v)\right)^{\alpha}$. For $0<\alpha<1$, we have the upper bound of $SO_{\alpha}(G)$ on unicyclic graphs with a fixed diameter, and the…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest…

Let $G$ be a graph on $n$ vertices and $m$ edges. For $\alpha \in [0,1]$, the $A_{\alpha}$-matrix of $G$ is defined as $A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G)$, where $A(G)$ is the adjacency matrix and $D(G)$ is the degree diagonal…

The eccentricity matrix of a simple connected graph G is obtained from the distance matrix of G by retaining the largest non-zero distance in each row and column, and the remaining entries are defined to be zero. A bi-block graph is a…