# Online talk: Lorenzo Traldi

##### Abstract:

Let $A=A(G)$ be the adjacency matrix of a simple graph $G$, considered as a matrix with entries in $GF(2)$. The binary matroid represented by the partitioned matrix $IAS(G)=\begin{pmatrix} I & A & A+I \end{pmatrix}$ is the isotropic matroid of $G$, denoted $M[IAS(G)]$. ($I$ is the identity matrix.) The matroid has three elements corresponding to each vertex of $G$.

Isotropic matroids have many interesting properties. One is the fact that two graphs are locally equivalent (up to isomorphism) if and only if their isotropic matroids are isomorphic. A special case underscores the difference between isotropic matroids and cycle matroids: two forests are isomorphic if and only if their isotropic matroids are isomorphic.

Another interesting property is that the isotropic matroid of $G$ contains two other kinds of structures associated with $G$, the delta-matroid and the isotropic system. Isotropic matroids allow us to use binary matroids to study the delta-matroids and isotropic systems of graphs.

After discussing these general properties, I’ll talk about joint work with Robert Brijder involving isotropic matroids of circle graphs. A circle graph $G$ is defined from an Euler circuit in a 4-regular graph $F$, and it turns out that in this case there is a precise relationship between the ranks of transversals in $M[IAS(G)]$ — a transversal is a subset that contains precisely one of the three matroid elements for each vertex — and the sizes of circuit partitions in $F$. This relationship is encoded in the often rediscovered circuit-nullity formula, which will celebrate the centennial of its first discovery next year (if it has not done so already). There are many different ways to characterize circle graphs using their isotropic matroids. The most striking of these characterizations is a multimatroid analogue of regularity: $G$ is a circle graph if and only if the ranks of the transversals of $M[IAS(G)]$ can be duplicated within a matroid representable over a field of characteristic other than 2.

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