Signed analogue of line graphs and their smallest eigenvalues

TL;DR

This paper extends Hoffman's theorem to signed graphs, showing that connected signed graphs with smallest eigenvalue greater than -2 and high minimum degree are essentially complete graphs, using new Hermitian matrix and signed graph concepts.

Contribution

It introduces Hoffman's limit theorem for Hermitian matrices and extends Hoffman graph and line graph concepts to signed graphs, providing a signed analogue of Hoffman's classical result.

Findings

Connected signed graphs with smallest eigenvalue > -2 and large minimum degree are switching equivalent to complete graphs.

Develops Hoffman's limit theorem for Hermitian matrices.

Extends Hoffman graph and line graph concepts to signed graphs.

Abstract

In this paper, we show that every connected signed graph with smallest eigenvalue strictly greater than $-2$ and large enough minimum degree is switching equivalent to a complete graph. This is a signed analogue of a theorem of Hoffman. The proof is based on what we call Hoffman's limit theorem which we formulate for Hermitian matrices, and also the extension of the concept of Hoffman graph and line graph for the setting of signed graphs.