Minimum Szeged index among unicyclic graphs with perfect matchings

Abstract

Let G be a connected graph. The Szeged index of G is defined as $$Sz(G)=\sum \nolimits _{e=uv\in E(G)}n_{u}(e|G)n_{v}(e|G)$$ , where $$n_{u}(e|G)$$ (resp., $$n_{v}(e|G)$$ ) is the number of vertices whose distance to vertex u (resp., v) is smaller than the distance to vertex v (resp., u), and $$n_{0}(e|G)$$ is the number of vertices equidistant from both ends of e. Let $$\mathcal {M}(2\beta )$$ be the set of unicyclic graphs with order $$2\beta $$ and a perfect matching. In this paper, we determine the minimum value of Szeged index and characterize the extremal graph with the minimum Szeged index among all unicyclic graphs with perfect matchings.