On graphs with eigenvectors in $\{1, -1, 0\}$ and the max $k$-cut problem

TL;DR

This paper characterizes graphs with eigenvectors having components in \\{-1, 0, 1\\} and extends the max k-cut problem to matrices, providing sharp bounds and constructing extremal graph families.

Contribution

It characterizes specific eigenvector structures and extends the max k-cut problem to matrices, establishing sharp bounds and extremal graph families.

Findings

Characterization of graphs with eigenvectors in \\{-1, 0, 1\\}.

Sharp upper bounds on max k-cut using eigenvalues.

Construction of infinite extremal graph families.

Abstract

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max $k$-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite families of extremal graphs for the obtained upper bounds.