On extremal cacti with respect to the Szeged index

The Szeged index of a graph G is defined as Sz(G)=∑e=uv∈Enu(e)nv(e), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u. A cactus is a graph in which any two cycles have at most one common vertex. Let C(n,k) denote the class of all cacti with order n and k cycles, and Cnt denote the class of all cacti with order n and t pendant vertices. In this paper, a lower bound of the Szeged index for cacti of order n with k cycles is determined, and all the graphs that achieve the lower bound are identified. As well, the unique graph in Cnt with minimum Szeged index is characterized.