Matching and symmetry of graphs

Abstract

Matching is a mathematical concept that deals with the way of spanning a given graph network with a set of pairs of adjacent points. It is pointed out that in many different areas of science and culture (e.g., physics, chemistry, games, etc.), computing the perfect and imperfect matching numbers is commonly performed but under different names, such as partition function for dimer statistics, Kekulé structures of molecules, paving domino problem. This paper demonstrate the mathematically beautiful but somewhat mystic relation between the symmetry of a graph and the factorable nature of its perfect matching number. There is introduced another interesting relation between the certain series of graphs and a family of orthogonal polynomials through the matching polynomial and topological index that are defined for counting the matching numbers.