A New Upper Bound for Laplacian Graph Eigenvalues

Let G= (V, E) be a graph with vertex set V = {v(1), v(2), ..., v(n)}. The degree of vi and the average degree of the adjacent vertices of vi are denoted by d(i) and m(vi), respectively. In this paper, we prove that max{d(v)m(v)/d(u) + d(u)m(u)/d(v) : uv is an element of E} is an upper bound for the largest Laplacian eigenvalue of G, the equality holds if G is a d-regular bipartite graph.