Bounds on graph eigenvalues I

We 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.