Cactus graphs with minimum edge revised Szeged index

The edge revised Szeged index Sze∗(G) is defined as Sze∗(G)=∑e=uv∈E(mu(e)+m0(e)∕2)(mv(e)+m0(e)∕2), where mu(e) and mv(e) are, respectively, the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u, and m0(e) is the number of edges equidistant to u and v. A cactus graph is a connected graph in which every block is an edge or a cycle. In this paper, we give a lower bound of the edge revised Szeged index among all m-edges cactus graphs with k cycles, and also characterize those graphs that achieve the lower bound. We also obtain the second minimum edge revised Szeged index for connected cactus graphs of size m with k cycles.