Related papers: On the Discrepancy Between Two Zagreb Indices

In a graph G, cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) is the (vertex) metric dimension of G. Similarly, the cardinality of such a set is the edge metric dimension of G, if it…

We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore,…

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when…

An extension of the well-known Szeged index was introduced recently, named as weighted Szeged index ($\textrm{sz}(G)$). This paper is devoted to characterizing the extremal trees and graphs of this new topological invariant. In particular,…

The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end…

For a graph $G$ with vertex set $V_{G}$ and edge set $E_{G}$, the symmetric division deg index is defined as $SDD(G)=\sum\limits_{uv\in E_{G}}(\frac{d_{u}}{d_{v}}+\frac{d_{v}}{d_{u}})$, where $d_{u}$ denotes the degree of vertex $u$ in $G$.…

For a graph G, M(G) denotes the maximum multiplicity occurring of an eigenvalue of a symmetric matrix whose zero-nonzero pattern is given by edges of G. We introduce two combinatorial graph parameters T^-(G) and T^+(G) that give a lower and…

Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let…

In a graph, we assign distinct integers to the vertices, and take the sum of two integers if they are on two adjacent vertices. The minimum possible number of different sums is the \emph{sum index} of this graph. In this paper, we present…

We investigate the \textit{edge group irregularity strength} ($es_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\mathcal{G}$ of order $s$, there exists a function $f:V(G)\rightarrow \mathcal{G}$ such…

The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $\chi'_{\Sigma}(G)$. It has been…

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension…

The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\{v\in V(G)\,|\,W(G)-W(G-v)=m \in \mathbb{Z} \}|/…

The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. We say vertices $u,v\in V$ are sum distinguished if $c(u)+\sum_{e\ni u}c(e)\neq c(v)+\sum_{e\ni v}c(e)$. By…

Let $c:V\cup E\to\{1,2,\ldots,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours,…

The $\sigma_{t}$-irregularity (or sigma total index) is a graph invariant which is defined as $\sigma_{t}(G)=\sum_{\{u,v\}\subseteq V(G)}(d(u)-d(v))^{2},$ where $d(z)$ denotes the degree of $z$. This irregularity measure was proposed by R\'…

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…

Let $G=(V,E)$ be a simple undirected and connected graph on $n$ vertices. The Graovac--Ghorbani index of a graph $G$ is defined as $$ABC_{GG}(G)= \sum_{uv \in E(G)} \sqrt{\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}},$$ where $n_u$ is the number of…

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…