Related papers: The average connectivity matrix of a graph

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…

We investigate the bounds on algebraic connectivity of graphs subject to constraints on the number of edges, vertices, and topology. We show that the algebraic connectivity for any tree on $n$ vertices and with maximum degree $d$ is bounded…

We investigate the ratio $\avM(G)$ of the average size of a maximal matching to the size of a maximum matching in a graph $G$. If many maximal matchings have a size close to $\maxM(G)$, this graph invariant has a value close to 1.…

A signed graph is a pair $(G,\Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V=\{1,\ldots,n\}$ and $\Sigma\subseteq E$. The edges in $\Sigma$ are called odd and the other edges of $E$…

For a graph property $\mathcal{P}$ and a common vertex set $V = \{1, 2, \ldots, n\}$, a family of graphs on $V$ is \emph{$\mathcal{P}$-intersecting} iff $G \cap H$ satisfies $\mathcal{P}$ for all $G,H$ in the family. Addressing a question…

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are…

Let $G=(V_G,E_G)$ be a connected graph. The distance $d_G(u,v)$ between vertices $u$ and $v$ in $G$ is the length of a shortest $u-v$ path in $G$. The eccentricity of a vertex $v$ in $G$ is the integer $e_G(v)= \max\{ d_G(v,u) \colon u\in…

The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the…

Let $S\subseteq V(G)$ and $\kappa_{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_1, T_2, \cdots, T_r$ in $G$ such that $V(T_i)\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots, r\}$ and $i\neq j$. For an integer $k$ with…

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…

For a graph $G$ of order $n$, the spectral sum of $G$ is defined to be the sum $\lambda_1(G) + \lambda_2(G)$, where $\lambda_1(G)$ (resp. $\lambda_2(G)$) is the largest (resp. second largest) adjacency eigenvalue of $G$. Ebrahimi, Mohar,…

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1].…

Let $G$ be a simple connected graph, and $D(G)$ be the distance matrix of $G$. Suppose that $D_{\max}(G)$ and $\lambda_1(G)$ are the maximum row sum and the spectral radius of $D(G)$, respectively. In this paper, we give a lower bound for…

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if…

We study the commuting graph of $n\times n$ matrices over the field of $p$-adics $\mathbb{Q}_p$, whose vertices are non-scalar $n\times n$ matrices with entries in $\mathbb{Q}_p$ and whose edges connect pairs of matrices that commute under…

The algebraic connectivity $a(G)$ of a graph $G$ is defined as the second smallest eigenvalue of its Laplacian matrix $L(G)$. It also admits a variational characterization as the minimum of a quadratic form associated with $L(G)$, subject…

A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the…

The \textit{longest path transversal number} of a connected graph $G$, denoted by $lpt(G)$, is the minimum size of a set of vertices of $G$ that intersects all longest paths in $G$. We present constant upper bounds for the longest path…

An obstacle representation of a graph $G$ is a set of points in the plane representing the vertices of $G$, together with a set of polygonal obstacles such that two vertices of $G$ are connected by an edge in $G$ if and only if the line…

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…