On the linear algebra of local complementation

TL;DR

This paper investigates the relationship between symmetric matrices over GF(2) and 4-regular graph circuit theory, revealing that local complementation equivalence can be characterized through inverse matrix operations.

Contribution

It introduces a novel linear algebraic perspective on local complementation, linking graph transformations to matrix operations over GF(2).

Findings

Local complementation equivalence can be generated by inverse matrix operations.

A new algebraic framework connects graph theory and matrix algebra over GF(2).

The approach offers insights into graph transformations and their algebraic representations.

Abstract

We explore the connections between the linear algebra of symmetric matrices over GF(2) and the circuit theory of 4-regular graphs. In particular, we show that the equivalence relation on simple graphs generated by local complementation can also be generated by an operation defined using inverse matrices.