The largest eigenvalue of unicyclic graphs

Let G   be a simple graph. Let λ1(G)λ1(G) and μ1(G)μ1(G) denote the largest eigenvalue of the adjacency matrix and the Laplacian matrix of G  , respectively. Let ΔΔ denote the largest vertex degree. If G has just one cycle, thenλ1(G)⩽2Δ-1.The equality holds if and only if G≅CnG≅Cn.Andμ1(G)⩽Δ+2Δ-1.The equality holds if and only if G≅CnG≅Cn, n is even.