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

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 $C^{\prime}$ are Euler systems of $F$ then $M(C,C^{\prime})$ and $M(C^{\prime},C)$ are inverses, and for any circuit partition $P$, $M(C^{\prime},P)=M(C^{\prime},C)\cdot M(C,P)$. This machinery allows for short proofs of several results regarding the linear algebra of interlacement

On the linear algebra of local complementation

AbstractWe 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