Integral trees with diameter four

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. In this paper, we investigate integral trees S(r;mi)=S(a1+a2+⋯+as;m1,m2,…,ms)S(r;mi)=S(a1+a2+⋯+as;m1,m2,…,ms) of diameter 4 with s=3,4,5,6s=3,4,5,6. Such integral trees are found by using a computer search or solving the Diophantine equations. New sufficient conditions for a construction of infinite families of integral trees S(r′;mi)=S(b1+⋯+bs;m1,S(r′;mi)=S(b1+⋯+bs;m1,…,…,ms  ) of diameter 4 from given integral trees S(r;mi)=S(a1+⋯+as;m1,S(r;mi)=S(a1+⋯+as;m1,…,…,ms  ) of diameter 4 are given. Further, using these conditions we construct infinitely many new classes of integral trees S(r′;mi)=S(b1+⋯+bs;m1,S(r′;mi)=S(b1+⋯+bs;m1,…,…,ms  ) of diameter 4 with s=3,4,5,6s=3,4,5,6. Finally, we propose two basic open problems about integral trees of diameter 4 for further study.