Max k-cut and the smallest eigenvalue

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 $\mathrm{mc}_{k}\left( G\right) \leq\frac{k-1}{k}\left( m-\frac{μ_{\min }\left( G\right) n}{2}\right) ,$ where $μ_{\min}\left( G\right)$ is the smallest eigenvalue of the adjacency matrix of $G.$ An infinite class of graphs forcing equality in this bound is constructed.

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