An isoperimetric constant for signed graphs

TL;DR

This paper introduces a signed Laplacian for graphs and defines an isoperimetric constant that bounds the first eigenvalue, motivated by graph lifts and higher-degree Laplacians.

Contribution

It presents a new signed Laplacian and an associated isoperimetric constant, extending spectral graph theory to signed graphs and higher-degree Laplacians.

Findings

Defines a signed Laplacian for graphs.

Establishes bounds for the first eigenvalue using the isoperimetric constant.

Motivated by graph lifts and higher-degree Laplacians.

Abstract

A sign is introduced in the usual Laplacian on graphs and the corresponding analogue of the isoperimetric constant for this Laplacian is presented, i.e. a geometric quantity which enables to bound from above and below the first eigenvalue. The introduction of the sign in the Laplacian is motivated by the study of $2$-lifts of graphs and of the combinatorial Laplacian in higher degree.