Integral trees with diameter 6

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. In this paper, two new classes of trees T(i,j)•T(p,q)•T(r,m,t) and K_1,s•T(i,j)•T(p,q)•T(r,m,t) of diameter 6 are defined. We obtain their characteristic polynomials and give the necessary and sufficient conditions for them to be integral. We also present some sufficient conditions of such trees to be integral by computer search. We also prove that the problem of finding integral trees of diameter 6 is equivalent to the problem of solving some Diophantine equations. Finally, we propose two basic open problems about integral trees of diameter 6 for further study.