Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter

Gutman and Wagner proposed the concept of matching energy (ME) and pointed out that the chemical applications of ME go back to the 1970s. Let $G$ be a simple graph of order $n$ and ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of ${\mu }_i(i=1,2,\dots ,n)$. In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter $d$.