The multiplicities of eigenvalues of a graph

Zhiwen Wang

Applied Mathematics and Computation, 2022, vol. 421, issue C

Abstract: 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.

Keywords: Graphs; Eigenvalues; Multiplicities (search for similar items in EconPapers)
Date: 2022
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:421:y:2022:i:c:s0096300322000431

DOI: 10.1016/j.amc.2022.126957

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