Related papers: On the difference between the eccentric connectivi…

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and…

Let $G=(V,E)$ be a finite simple graph. The Graovac-Ghorbani index of a graph G is defined as $ABC_{GG}(G)=\sum_{uv\in E(G)}\sqrt{\frac{n_u(uv,G)+n_v(uv,G)-2}{n_u(uv,G)n_v(uv,G)}},$ where $n_u(uv,G)$ is the number of vertices closer to…

The eccentricity (anti-adjacency) matrix $\varepsilon(G)$ of a graph $G$ is obtained from the distance matrix by retaining the eccentricities in each row and each column. The $\varepsilon$-eigenvalues of a graph $G$ are those of its…

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the…

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with connectivity at least $c$, and…

The distance of a vertex in a graph is the sum of distances from that vertex to all other vertices of the graph. The Wiener index of a graph is the sum of distances between all its unordered pairs of vertices. A graph has been obtained that…

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…

Let $G=(V,E)$ be a simple graph. The concept of Inverse symmetric division deg index $(ISDD)$ was introduced in the chemical graph theory very recently. In spite of this, a few papers have already appeared with this index in the literature.…

Graph rigidity, the study of vertex realizations in $\mathbb{R}^d$ and the motions that preserve the induced edge lengths, has been the focus of extensive research for decades. Its equivalency to graph connectivity for $d=1$ is well known;…

The Randi\'{c} index $R(G)$ of a graph $G$ is defined as the sum of $(d_i d_j)^{-1/2}$ over all edges $v_i v_j$ of $G$, where $d_i$ is the degree of the vertex $v_i$ in $G$. The radius $r(G)$ of a graph $G$ is the minimum graph eccentricity…

Given a connected graph $G(V, E)$, the edge dimension, denoted $\mathrm{edim}(G)$, is the least size of a set $S \subseteq V$ that distinguishes every pair of edges of $G$, in the sense that the edges have pairwise distinct tuples of…

Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{\delta+1}+2\big) {n…

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two…

For a graph $G=(V,E)$, its exact-distance square, $G^{[\sharp 2]}$, is the graph with vertex set $V$ and with an edge between vertices $x$ and $y$ if and only if $x$ and $y$ have distance (exactly) $2$ in $G$. The graph $G$ is an…

The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers…

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…

A vertex $v$ of a connected graph $G$ is said to be a boundary vertex of $G$ if for some other vertex $u$ of $G$, no neighbor of $v$ is further away from $u$ than $v$. The boundary $\partial(G)$ of $G$ is the set of all of its boundary…

Let \( D \) be a strongly connected digraph. The average distance of a vertex \( v \) in \( D \) is defined as the arithmetic mean of the distances from \( v \) to all other vertices in \( D \). The remoteness \( \rho(D) \) of \( D \) is…

A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $\kappa'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components…

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$…