On Extremal Cacti with Respect to the Edge Szeged Index and Edge-vertex Szeged Index

The edge Szeged index and edge-vertex Szeged index of a graph are defined as Sz(e)(G) = Sigma(uv is an element of E(G) )m(u) (uv vertical bar G)m(v)(uv vertical bar G) and Sz(ev)(G) = 1/2 Sigma(uv is an element of E(G)) [n(u)(uv vertical bar G)m(v)(uv vertical bar G) + n(v)(uv vertical bar G)m(u)(uv vertical bar G)], respectively, where m(u)(uv vertical bar G) (resp., m(v)(uv vertical bar G)) and n(u) (uv vertical bar G) (resp., n(v)(uv vertical bar G)) are the number of edges and vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u), respectively. A cactus is a graph in which any two cycles have at most one common vertex. In this paper, the lower bounds of edge Szeged index and edge-vertex Szeged index for cacti with order n and k cycles are determined, and all the graphs that achieve the lower bounds are identified.