Cacti with the smallest,second smallest,and third smallest Gutman index

The Gutman index (also known as Schultz index of the second kind) of a graph \(G\) is defined as \(Gut(G)=\sum \nolimits _{u,v\in V(G)}d(u)d(v)d(u, v)\). A graph \(G\) is called a cactus if each block of \(G\) is either an edge or a cycle. Denote by \(\mathcal {C}(n, k)\) the set of connected cacti possessing \(n\) vertices and \(k\) cycles. In this paper, we give the first three smallest Gutman indices among graphs in \(\mathcal {C}(n, k)\), the corresponding extremal graphs are characterized as well.