Max k-cut and the smallest eigenvalue

TL;DR

This paper establishes an upper bound for the maximum size of a k-cut in a graph using its smallest eigenvalue and constructs graphs that achieve this bound, linking spectral properties to combinatorial optimization.

Contribution

It introduces a new spectral bound for the maximum k-cut size and constructs graphs that attain this bound, advancing understanding of eigenvalue-based graph partitioning.

Findings

Derived an upper bound for max k-cut using the smallest eigenvalue.

Constructed an infinite class of graphs where the bound is tight.

Linked spectral graph theory with combinatorial optimization.

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.