Abstract
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 ρ( G) of G satisfies ρ ( G ) ⩽ 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 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.
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