An integral tree is a tree whose adjacency matrix has only integer eigenvalues. While most previous work by other authors has been focused either on the very restricted case of balanced trees or on finding trees with diameter as large as possible, we study integral trees of diameter 4. In particular, we characterize all diameter 4 integral trees of the form T(m₁, t₁) • T(m₂, t₂). In addition we give elegant parametric descriptions of infinite families of integral trees of the form T(m₁, t₁) • · · · • T(mn, tn) for any n > 1. We conjecture that we have found all such trees.
AKCE International Journal of Graphs and Combinatorics Vol. 7, Issue 2, p. 171-188