Maximum Zagreb Indices Among All $p-$Quasi $k-$Cyclic Graphs

Abstract

vspace{0.2cm}<br />Suppose $G$ is a simple and connected graph. The first and second Zagreb indices of $G$ are two degree-based graph invariants defined as $M_1(G) = sum_{v in V(G)}deg(v)^2$ and $M_2(G) = sum_{e=uv in E(G)}deg(u)deg(v)$, respectively. The graph $G$ is called $p-$quasi $k-$cyclic, if there exists a subset $S$ of vertices such that $|S| = p$, $G setminus S$ is $k-$cyclic and there is no a subset $S^prime$ of $V(G)$ such that $|S^prime| < |S|$ and $G setminus S^prime$ is $k-$cyclic. The aim of this paper is to characterize all graphs with maximum values of Zagreb indices among all $p-$quasi $k-$cyclic graphs with $k leq 3$. & & vspace{0.2cm}