The construction of graphs with irreducible matching polynomials and their generalizations

This paper investigates methods for constructing graphs whose matching polynomials are irreducible over [Formula: see text]. Building on this, the construction method is extended to general graph polynomials, for graphs whose polynomials satisfy certain properties, the connection construction produces a series of irreducible polynomial, such as characteristic polynomials and matching polynomials. Finally, the paper presents algorithms for calculating the matching polynomial of a given graph and the matching polynomial for a given number of vertices, implemented in Python.