On the reduced second Zagreb index of graphs

Abstract

For a connected graph $G$, the reduced second Zagreb index is defined as

$$R{M}_2=R{M}_2\left(G\right)=\sum _{uv\in E\left(G\right)}\left(d_G\left(u\right)-1\right)\left(d_G\left(v\right)-1\right),$$

where $d_G\left(u\right)$ and $d_G\left(v\right)$ denote the degrees of vertices $u$ and $v$ in graph $G$, respectively. We determine the extremal graphs among all connected graphs of order $n$ and with $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)-m$ edges with respect to $R{M}_2$, and obtain the first ten largest $R{M}_2$ for connected graphs of order $n$. Moreover, we characterize the sharp upper bound on $R{M}_2$ for all connected bipartite graphs with given connectivity $k$. Finally, the extremal graphs with respect to $R{M}_2$ are determined in all connected bipartite graphs with matching number $q$.