A characterization of trees with eigenvalue multiplicity one less than their number of pendant vertices

Abstract

The multiplicity of an eigenvalue λ of a connected graph G is denoted by m(G,λ). Denote by θ(G)=|E(G)|−|V(G)|+1 the cyclomatic number of G and by p(G) the number of pendant vertices of G. In 2020, Wang et al. [Linear Algebra Appl. 584(2020)] gave an upper bound for the eigenvalue multiplicity of a graph G in terms of the number of pendant vertices and the cyclomatic number of G. The authors proved that if a connected graph G is not a cycle, then m(G,λ)≤2θ(G)+p(G)−1, and left a problem: To characterize all the connected graphs G together with the eigenvalue λ such thatm(G,λ)=2θ(G)+p(G)−1. It was especially mentioned that: Even if G is assumed to be a tree, this problem is worth of studying. In the present article, we solve the problem for the case when G is a tree. It is proved that if T is a tree with m(T,λ)=p(T)−1, then there are two positive integers m and k such that λ=2coskπm+1,1≤k≤m. Moreover, a characterization of all trees T with m(T,λ)=p(T)−1 is given.