Bounds on eigenvalues and chromatic numbers

We give new bounds on eigenvalue of graphs which imply some known bounds. In particular, if T(G) is the maximum sum of degrees of vertices adjacent to a vertex in a graph G, the largest eigenvalue rho(G) of G satisfies rho(G) less than or equal to root T(G) with equality if and only if either G is regular or G is bipartite and such that all vertices in the same part have the same degree. Consequently, we prove that the chromatic number of G is at most root T(G) + 1 with equality if and only if G is an odd cycle or a complete graph, which implies Brook's theorem. A generalization of this result is also given. (C) 1998 Elsevier Science Inc.