Extremal $k$-generalized quasi unicyclic graphs with respect to first and second Zagreb indices

Abstract

The first and second Zagreb indices of a graph $G$ are defined as: $M_1\left(G\right)={\sum }_{v\in V\left(G\right)}d{\left(v\right)}^2$ and $M_2\left(G\right)={\sum }_{uv\in E\left(G\right)}d\left(u\right)d\left(v\right)$, where $d\left(v\right)$ is the degree of the vertex $v$. A graph $G$ is said to be a quasi unicyclic graph, if there exists a vertex $z\in V\left(G\right)$, such that $G-z$ is a unicyclic graph and $z$ is called a quasi vertex. For any integer $k\ge 1$ a graph $G$ is called a $k$-generalized quasi unicyclic graph, if there exists a subset $V_k\subseteq V\left(G\right)$ with cardinality $k$ such that $G-V_k$ is a unicyclic graph but for every subset $V_{k-1}$ of cardinality $k-1$ of $V\left(G\right)$, the graph $G-V_{k-1}$ is not unicyclic graph. In this paper, we investigate the upper and lower bounds on first and second Zagreb indices for $k$-generalized quasi unicyclic graphs. Moreover, we characterize the extremal graphs with maximum and minimum Zagreb indices.