New Classes of Integral Trees of Diameter $4$

Abstract

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral trees $S(r;m_i)=S(a_1+a_2+\cdots +a_s;m_1,m_2,\dots ,m_s)$ of diameter $4$ with $s=2,3$. We give a better sufficient and necessary condition for the tree $S(a_1+a_2;m_1,m_2)$ of diameter $4$ to be integral, from which we construct infinitely many new classes of such integral trees by solving some certain Diophantine equations. These results are different from those in the existing literature. We also construct new integral trees $S(a_1+a_2+a_3;m_1,m_2,m_3)=S(a_1+1+1;m_1,m_2,m_3)$ of diameter $4$ with non-square numbers $m_2$ and $m_3$. These results generalize some well-known results of P.Z. Yuan, D.L. Zhang $et$ $al$.