ABSTRACT
This chapter provides a preparation for the definition of the concept of a matching polynomial. Here, we give some basic definitions and recall a few necessary facts from spectral graph theory. In the mathematical literature, the concept of the matching polynomial was introduced by Edward Farrell]. This paper appeared in 1979 but was received by the journal already in 1966. Farrell considered the counting polynomial of the matching numbers. In the 1970s, the matching polynomial emerged also in theoretical chemistry within a theoretical model of aromaticity and resonance energy. The concept of matching is a classical topic of graph theory. Countless books, surveys, and research papers are devoted to matchings in graphs. A legion of results exists in the mathematical and chemical literature on matching polynomials of special classes of graphs. The matching polynomial belongs among the very few combinatorial graph polynomials that have deep-lying algebraic properties.
