The Second-minimum Gutman Index of The Unicyclic Graphs With Given Girth

Abstract

Let $G$ be a simple connected graph with vertex set $V(G)$. The Gutman index $\text{Gut}(G)$ of $G$ is defined as $\text{Gut}(G)=\sum _{\{x,y\}\subseteq V(G)}{d}_G(x)d_G(y)d_G(x,y)$, where $d_G(x)$ is the degree of vertex $v$ in $G$ and $d_G(x,y)$ is the distance between vertices $x$ and $y$ in $G$. In this paper, the second-minimum Gutman index of unicyclic graphs on $n$ vertices and girth $m$ is characterized.