For a graph $G$, $\chi (G) = 1+ \lambda_1$ iff $G$ be a complete or oddcycle connected graph. For a graph $G$, $\chi (G)$ is the color number. Means minimum colors need to paint vertexes of graph whit out any couple of vertexes whit the same color. $\lambda_1$ Is the greatest eigenvalue of adjacency matrix of graph. I need some idea or an answer for it. Thank for reading.

This is a theorem due to Herb Wilf, google on "wilf chromatic number". The paper is only two pages (and a few lines) long.