Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices

TL;DR

This paper introduces a novel method for modifying interlacement matrices in 4-regular graphs using nonsymmetric matrices, simplifying proofs of linear algebraic properties related to circuit partitions.

Contribution

It presents a new approach to interlacement matrices that enables easier analysis and proofs of properties in 4-regular graphs using nonsymmetric matrices.

Findings

M(C,C') and M(C',C) are inverses for Euler systems C and C'

M(C',P)=M(C',C)M(C,P) for any circuit partition P

Simplifies proofs of linear algebra results in interlacement theory

Abstract

Let F be a 4-regular graph with an Euler system C. We introduce a simple way to modify the interlacement matrix of C so that every circuit partition P of F has an associated modified interlacement matrix M(C,P). If C and C' are Euler systems of F then M(C,C') and M(C',C) are inverses, and for any circuit partition P, M(C',P)=M(C',C)M(C,P). This machinery allows for short proofs of several results regarding the linear algebra of interlacement.