Abstract

A graph G is called integral if all eigenvalues of its adjacency matrix A(G) are integers. In this paper, the trees T(p,q)•T(r,m,t) and K_1,s•T(p,q)•T(r,m,t) of diameter 6 are defined. We determine their characteristic polynomials. We also obtain for the first time sufficient and conditions for them to be integral. To do so, we use number theory and apply a computer search. New families of integral trees of diameter 6 are presented. Some of these classes are infinite. They are different from those in the existing literature. We also prove that the problem of finding integral trees of diameter 6 is equivalent to the problem of solving some Diophantine equations. We give a positive answer to a question of Wang et al. [Families of integral trees with diameters 4, 6 and 8, Discrete Appl. Math. 136 (2004) 349–362].

Keywords: Integral tree; Characteristic polynomial; Graph spectrum

Mathematical subject codes: 05C50; 05C05; 11D09; 11D41

Article Outline

1. Introduction

2. The characteristic polynomials of two classes of trees

3. Integral trees of diameter 6

Acknowledgements

References

Supported by National Science Foundation of China and the fund of the Developing Program for Outstanding Persons in NPU.