34 research outputs found
Nullity Invariance for Pivot and the Interlace Polynomial
We show that the effect of principal pivot transform on the nullity values of
the principal submatrices of a given (square) matrix is described by the
symmetric difference operator (for sets). We consider its consequences for
graphs, and in particular generalize the recursive relation of the interlace
polynomial and simplify its proof.Comment: small revision of Section 8 w.r.t. v2, 14 pages, 6 figure
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
The adjacency matroid of a graph
If is a looped graph, then its adjacency matrix represents a binary
matroid on . may be obtained from the delta-matroid
represented by the adjacency matrix of , but is less sensitive to
the structure of . Jaeger proved that every binary matroid is for
some [Ann. Discrete Math. 17 (1983), 371-376].
The relationship between the matroidal structure of and the
graphical structure of has many interesting features. For instance, the
matroid minors and are both of the form
where may be obtained from using local
complementation. In addition, matroidal considerations lead to a principal
vertex tripartition, distinct from the principal edge tripartition of
Rosenstiehl and Read [Ann. Discrete Math. 3 (1978), 195-226]. Several of these
results are given two very different proofs, the first involving linear algebra
and the second involving set systems or delta-matroids. Also, the Tutte
polynomials of the adjacency matroids of and its full subgraphs are closely
connected to the interlace polynomial of Arratia, Bollob\'{a}s and Sorkin
[Combinatorica 24 (2004), 567-584].Comment: v1: 19 pages, 1 figure. v2: 20 pages, 1 figure. v3:29 pages, no
figures. v3 includes an account of the relationship between the adjacency
matroid of a graph and the delta-matroid of a graph. v4: 30 pages, 1 figure.
v5: 31 pages, 1 figure. v6: 38 pages, 3 figures. v6 includes a discussion of
the duality between graphic matroids and adjacency matroids of looped circle
graph
Interlace Polynomials for Multimatroids and Delta-Matroids
We provide a unified framework in which the interlace polynomial and several
related graph polynomials are defined more generally for multimatroids and
delta-matroids. Using combinatorial properties of multimatroids rather than
graph-theoretical arguments, we find that various known results about these
polynomials, including their recursive relations, are both more efficiently and
more generally obtained. In addition, we obtain several interrelationships and
results for polynomials on multimatroids and delta-matroids that correspond to
new interrelationships and results for the corresponding graphs polynomials. As
a tool we prove the equivalence of tight 3-matroids and delta-matroids closed
under the operations of twist and loop complementation, called vf-safe
delta-matroids. This result is of independent interest and related to the
equivalence between tight 2-matroids and even delta-matroids observed by
Bouchet.Comment: 35 pages, 3 figure