Note on graphs with irreducible characteristic polynomials

Abstract

Let G be a connected simple graph with characteristic polynomial PG(x). The irreducibility of PG(x) over rational numbers Q has a close relationship with the automorphism group, reconstruction and controllability of a graph. In this paper we derive three methods to construct graphs with irreducible characteristic polynomials by appending paths P2n+1−2(n≥1) to certain vertices; union and join K1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results.