2025/07/21 by Gupta, Vinayak, Lather, Gargi, Balaji, R. · 1 citation
#05C05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.15472
Let T be a tree. Suppose λ is an eigenvalue of the Laplacian matrix of T with multiplicity mT(λ). It is known that mT(λ) ≤ p(T)-1, where p(T) is the number of pendant vertices of T. In this paper, we characterize all trees T for which there exists an eigenvalue λ such that mT(λ)=p(T)-1. We show that such trees are precisely either paths, or there exists an integer q such that if α and β are two distinct pendant vertices, then the distance d(α,β) satisfies d(α, β) ≡ 2q ~\rmmod~(2q+1). As a consequence, we show that 1 is an eigenvalue of LT with multiplicity p(T)-1 if and only if d(α,β) ≡ 2 mod 3 for all distinct pendant vertices α and β of T.