关键词: 匹配多项式, 匹配能量, 匹配数, 树, 单圈图

Abstract: The concept of graph energy originates from chemistry and physics, particularly organic chemistry and statistical physics. It was first introduced by Erich Hückel in 1978 through computing the total π-electron energy. Graph energy is a parameter defined on the molecular structure of matter, which is closely related to the molecular structure, and scientists use it as an approximation of the Hückel molecular orbit. In physics, HEILMANN and LIEB (1970) first proposed the concept of matching roots of molecular structural graphs. In 1972, they studied the matching roots of molecular structural graphs in their paper “Theory of Monomer-Dimer Systems”. They related the monomer-dimer problem to the Heisenberg and Ising models of magnetism and derived Christoffel-Darboux formulas for the partition functions of these models. They also proved that the roots of the matching polynomial of graphs are real, and computed the expression for a certain type of graph. At around the same time, mathematicians such as Godsil and Gutman independently proved this result by different approaches. Subsequently, more mathematicians began to explore the combinatorial properties of graph parameters related to matching roots, matching numbers, the number of perfect matchings, and matching energies of graphs.
GUTMAN and WAGNER (2012) introduced the concept of matching energy, defined as the sum of the absolute values of the roots of matching polynomial of a graph. They presented basic results for simple graphs and studied the matching energy of trees, unicyclic graphs, complete bipartite graphs, and approximation upper bound for the complete graphs.Their work laid the foundation for further research, leading to numerous publications on the matching energy of various classes of simple graphs. Researchers have calculated the largest matching energy and the minimum matching energy, the largest matching roots for trees, unicyclic, bicyclic, and tricyclic graphs, and “Unmarked” characterized the extremal graphs with the minimum or maximum matching energy.
Graph matching energy, Hosoya index, the largest matching root (also known as the matching index), and matching number are highly related parameters, though quantifying their exact relationships remains challenging. JI et al. (2013) studied the extremal matching energy of bicyclic graphs. ZHU (2013) also investigated the minimal energies of unicyclic graphs with perfect matchings, identifying the first seven minimal energies and addressing a conjecture. WANG and SO (2015) explored the minimum matching energy of graphs, examining the relationship between matching energy and several graph transformations. Most studies in this area involve computing the matching energies of graphs within specific classes and characterizing the corresponding extremal graphs. ZHANG and LIU (2021) studied the behavior of the largest matching root and the matching energy of graphs under two graph transformations. The results were published in the MATCH Communications in Mathematical and in Computer Chemistry in 2021.
In this paper, we investigate how the coefficients and the largest matching root of matching polynomials of graphs behave under two specific graph transformations. The result shows that the graph has the largest matching root but the minimum matching energy. The largest Hosoya index also induces the largest matching energy. As an application, we characterize the graphs with minimum matching energy among all trees and unicyclic graphs with a given matching number. We also provide numerical lower bounds for the matching energy of these graphs in terms of their matching number.

Key words: matching polynomial, matching energy, matching number, tree, unicycle graph