The multiplicities of eigenvalues of a graph

For a connected graph G, let e(G) be the number of its distinct eigenvalues and d be the diameter. It is well known that e(G)≥d+1. This shows η≤n−d, where η and n denote the nullity and the order of G, respectively. A graph is called minimal if e(G)=d+1. In this paper, we characterize all trees satisfying η(T)=n−d or n−d−1. Applying this result, we prove that a caterpillar is minimal if and only if it is a path or an even caterpillar, which extends a result by Aouchiche and Hansen. Furthermore, we completely characterize all connected graphs satisfying η=n−d. For any non-zero eigenvalue of a tree, a sharp upper bound of its multiplicity involving the matching number and the diameter is provided.