A characterization on trees T with m(T, λ)=p(T)-2

Let m(G,λ) be the multiplicity of an eigenvalue λ of a connected graph G. Wang et al. [Linear Algebra Appl. 584(2020), 257-266] proved that for any connected graph G≠ Cₙ, m(G, λ) ≤ 2c(G) + p(G) -1, where $c (G) = |E(G)| - |V (G)| + 1$ and $p(G)$ are the cyclomatic number and the number of pendant vertices of G, respectively. In the same paper, they proposed the problem to characterize all connected graphs G with eigenvalue λ such that m(G, λ) =2c (G)+ p(G)-1. Wong et al. [Discrete Math. 347(2024), 113845] solved this problem for the case when G is a tree by characterizing all trees T with eigenvalue λ such that m(T , λ) = p(T )-1. In this paper, we further provide the structural characterization on trees T with eigenvalue λ such that m(T , λ) = p(T )-2.