Bounds on graph eigenvalues I

AbstractWe improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n⩾2, maximum degree Δ, and girth at least 5, thenμ(G)⩽mina,n-1},where μ(G) is the largest eigenvalue of the adjacency matrix of G.Also, if G is a graph of order n⩾2 with dominating number γ(G)=γ, thenλ2(G)⩽nifγ=1,n-γifγ⩾2,λn(G)⩾⌈n/γ⌉,where 0=λ1(G)⩽λ2(G)⩽⋯⩽λn(G) are the eigenvalues of the Laplacian of G.We also determine all cases of equality in the above inequalities