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Gutman and Wagner proposed the concept of matching energy (ME) and pointedout that the chemical applications of ME go back to the 1970s. Let $G$ be asimple graph of order $n$ and $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of itsmatching polynomial. The matching energy of $G$ is defined to be the sum of theabsolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. In this paper, we characterizethe graphs with minimal matching energy among all unicyclic and bicyclic graphswith a given diameter $d$.

Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter