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 \(\text{Gut}(G) = \sum\limits_{\{x,y\} \subseteq V(G)} d_G(x) d_G(y) d_G(x,y)\), 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.