On the first and second Zagreb indices of quasi unicyclic graphs

Abstract

‎Let $G$ be a simple graph‎. ‎The graph $G$ is called a quasi unicyclic graph if there exists a vertex $x \in V(G)$ such that $G-x$ is a connected graph with a unique cycle‎. ‎Moreover‎, ‎the first and the second Zagreb indices of $G$ denoted by $M_1(G)$ and $M_2(G)$‎, ‎are the sum of $\deg^2(u)$ overall vertices $u$ in $G$ and the sum of $\deg(u)\deg(v)$ of all edges $uv$ of $G$‎, ‎respectively‎. ‎The first and the second Zagreb indices are defined relative to the degree of vertices‎. ‎In this paper‎, ‎sharp upper and lower bounds for the first and the second Zagreb indices of quasi unicyclic graphs are given‎.

Keywords

• ‎first zagreb index‎
• second zagreb index‎
• quasi unicyclic graphs