Bounds of eigenvalues of a nontrivial bipartite graph

摘要

Let G be a simple graph with n vertices and m edges, and let λ_1 and λ_2 denote the largest and second largest eigenvalues of G. For a nontrivial bipartite graph G, we prove that, (i) λ_1 ≤,where equality holds if and only if G ≌ P_4; (ii) If G (≈≠) P_n, then, where equality holds if and only if G ≌ K_(2,3)- e; (iii) If G is connected, then, where equality holds if and only if G ≌ P_n, 2 ≤ n ≤ 5; (iv) λ_2,where equality holds if and only if G ≌ P_4; (v) If G is connected and G (≈≠) P_n,then λ_2 > equality holds if and only if G ≌ K_(2,3) - e.