Note on graphs with irreducible characteristic polynomials.

Let G be a connected simple graph with characteristic polynomial P G (x). The irreducibility of P G (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 P 2 n + 1 − 2 (n ≥ 1) to certain vertices; union and join K 1 alternately and corona. These methods are based on Eisenstein's criterion and field extensions. Concrete examples are also supplied to illustrate our results. [ABSTRACT FROM AUTHOR]