The Maximal Matching Energy of Tricyclic Graphs

TL;DR

This paper characterizes and determines the tricyclic graphs with the highest matching energy, using Coulson-type integral formulas and extending prior computational and inductive results.

Contribution

It provides a complete characterization of tricyclic graphs with maximal matching energy, advancing understanding of spectral properties in chemical graph theory.

Findings

Identifies the graphs with maximal matching energy among tricyclic graphs.

Uses Coulson-type integral formula to compare matching energies.

Completes the classification of extremal tricyclic graphs.

Abstract

Gutman and Wagner proposed the concept of the 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 $\mu_{i}\ (i=1,2,\ldots,n)$. Gutman and Cvetkoi\'c determined the tricyclic graphs on $n$ vertices with maximal number of matchings by a computer search for small values of $n$ and by an induction argument for the rest. Based on this result, in this paper, we characterize the graphs with the maximal value of matching energy among all tricyclic graphs, and completely determine the tricyclic graphs with the maximal matching energy. We prove our result by using Coulson-type integral formula of matching energy, which is similar as the method to comparing the energies of two quasi-order incomparable graphs.