Integral trees of odd diameters

TL;DR

This paper proves that for every odd diameter greater than 1, there exist infinitely many integral trees with that diameter, expanding the known classes of such graphs.

Contribution

It establishes the existence of infinitely many integral trees for every odd diameter greater than 1, filling a gap in the classification of integral trees.

Findings

Existence of infinitely many integral trees for each odd diameter n > 1

Extension of previous results limited to small diameters

Completes the characterization of integral trees by diameter

Abstract

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. Recently, Csikvari proved the existence of integral trees of any even diameter. In the odd case, integral trees have been constructed with diameter at most 7. In this paper, we show that for every odd integer n > 1, there are infinitely many integral trees of diameter n.