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

Let m (G, ) be the multiplicity of an eigenvalue 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, ) 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 such that m (G, ) =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 such that m (T, ) = p (T) -1. In this paper, we further provide the structural characterization on trees T with eigenvalue such that m (T, ) = p (T) -2.