Unimodularity and circle graphs

AbstractA property of unimodularity is introduced for antisymmetric integral matrices. It is satisfied by the adjacency matrix of a circle graph provided with a Naji orientation [8]. In a further paper we shall interprete this result in terms of symmetric matroids introduced in [2]. In this communication we give a direct proof by means of techniques used in [1] for an algorithmic solution of the Gauss problem on self-intersecting surves in the plane

A note on the spectra of certain skew-symmetric {1,0,−1}-matrices

AbstractWe characterize skew-symmetric {1,0,−1}-matrices with a certain combinatorial property. In particular, we exhibit several equivalent descriptions of this property. These results allow characterizations of unimodular orientations of the complete graph, of rank 2 chirotopes, and of a class of multipartite oriented graphs

The Interlace Polynomial

In this paper, we survey results regarding the interlace polynomial of a graph, connections to such graph polynomials as the Martin and Tutte polynomials, and generalizations to the realms of isotropic systems and delta-matroids.Comment: 18 pages, 5 figures, to appear as a chapter in: Graph Polynomials, edited by M. Dehmer et al., CRC Press/Taylor & Francis Group, LL

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