Matching Energy of Unicyclic and Bicyclic Graphs with a Given Diameter

TL;DR

This paper characterizes unicyclic and bicyclic graphs with a fixed diameter that have the minimal matching energy, a graph invariant related to chemical applications and spectral properties.

Contribution

It provides a complete characterization of graphs with minimal matching energy among unicyclic and bicyclic graphs for a given diameter.

Findings

Identifies graphs with minimal matching energy for specified diameter.

Extends understanding of spectral graph invariants in chemical graph theory.

Offers potential applications in molecular stability analysis.

Abstract

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 $\mu_1,\mu_2,\ldots,\mu_n$ be the roots of its matching polynomial. The matching energy of $G$ is defined to be the sum of the absolute values of $\mu_{i}\ (i=1,2,\ldots,n)$. In this paper, we characterize the graphs with minimal matching energy among all unicyclic and bicyclic graphs with a given diameter $d$.