Max k-cut and the smallest eigenvalue

Abstract

Let $G$ be a graph of order $n$ and size $m$, and let $\mathrm{mc}_{k}\left( G\right)$ be the maximum size of a $k$-cut of $G.$ It is shown that $$\mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{\mu_{\min }\left( G\right) n}{2}\right) ,$$ where $\mu_{\min}\left( G\right)$ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.Comment: 5 pages. Some typos corrected in v