Trees with exactly three main eigenvalues

An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector.

Introduced by Cvetković in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood.

However, the classification of graphs with precisely three main eigenvalues remains a challenging open problem in spectral graph theory.

This paper provides a complete classification of all trees of diameter 5 with exactly three main eigenvalues.

Using equitable partitions, the spectral condition reduces to the unique solvability of linear systems over the rationals, leading to Diophantine equations involving branch lengths and pendant counts.

We prove that every such tree is isomorphic either to a symmetric tree $T_r(a)$ or to a member of a parametric family $\mathcal{T}$ determined by arithmetic divisibility conditions.

We also construct an infinite family of such trees with unbounded diameter.