Integral trees of diameter 6

A graph G   is called integral if all eigenvalues of its adjacency matrix A(G)A(G) are integers. In this paper, the trees T(p,q)•T(r,m,t)T(p,q)•T(r,m,t) and K1,s•T(p,q)•T(r,m,t)K1,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].