The minimum matching energy of bicyclic graphs with given girth

Abstract

The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. Let $\theta (r,s,t)$ be the graph obtained by fusing two triples of pendant vertices of three paths $P_{r+2}$, $P_{s+2}$ and $P_{t+2}$ to two vertices. The graph obtained by identifying the center of the star $S_{n-g}$ with the degree~3 vertex $u$ of $\theta (1,g-3,1)$ is denoted by $S_{n-g}(u)\theta (1,g-3,1)$. In this paper, we show that, $S_{n-g}(u)\theta (1,g-3,1)$ has minimum matching energy among all bicyclic graphs with order $n$ and girth $g$.