Bounds on graph eigenvalues I Original Research Article

We improve some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n greater-or-equal, slanted 2, maximum degree Δ, and girth at least 5, thenimagewhere μ(G) is the largest eigenvalue of the adjacency matrix of G. Also, if G is a graph of order n greater-or-equal, slanted 2 with dominating number γ(G) = γ, thenimagewhere 0 = λ1(G) less-than-or-equals, slant λ2(G) less-than-or-equals, slant cdots, three dots, centered less-than-or-equals, slant λn(G) are the eigenvalues of the Laplacian of G. We also determine all cases of equality in the above inequalities.