Bounds for Two Classes of Graph Spectrum and Proof of Their Conjectures

Let λ_k be the kth greatest eigenvalue of forest F (or tree T) on n vertices, then λ_k=-λ_(n-k 1). Hong Yuan proposed the following conjecture: Conjecture 1. Suppose T is a tree with n vertices and edge independence number q. For k≤q, λ_k(T)≥λ_k(S_(n-2k 2)~(2h-2) with equality iff T≌S_(n-2k 2)~(2k-2), where S_(n-2k 2)~(2k-2) is formed from a K_(1,n-2k 1) and a path P_(2k-2) by joining with an edge a vertex of degree one of P_(2k-2) to the vertex of degree n-2k 1 of K_(1,n-2k 1).