We introduce a new graph polynomial in two variables. This ``interlace'' polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the significant differences are that our expansion is over vertex rather than edge subsets, and the rank and nullity employed are those of an adjacency matrix rather than an incidence matrix.
The second computation is by a three-term reduction formula involving a graph pivot; the pivot arose previously in the study of interlacement and Euler circuits in four-regular graphs.
We consider a few properties and specializations of the two-variable interlace polynomial. One specialization, the ``vertex-nullity interlace polynomial'', is the single-variable interlace graph polynomial we studied previously, closely related to the Tutte-Martin polynomial on isotropic systems previously considered by Bouchet. Another, the ``vertex-rank interlace polynomial'', is equally interesting. Yet another specialization of the two-variable polynomial is the independent-set polynomial.
In this work, we study the interlace polynomial as a generalization of a graph invariant to delta-matroids. We prove that the interlace polynomial satisfies the four-term relation for delta-matroids and determines thus a finite type invariant of links in the 3-sphere.
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.
Let G be a graph with adjacency matrix A(G). Consider the matrix IA(G)=(I | A(G)), where I is the identity matrix, and let M(IA(G)) be the binary matroid represented by IA(G). Then suitably parametrized versions of the Tutte polynomial of M(IA(G)) yield the interlace polynomials of G, introduced by Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584]. Interlace polynomials subsequently introduced by other authors may be obtained from parametrized Tutte polynomials of the binary matroid represented by (I | A(G) | I+A(G)).
The generating function that records the sizes of directed circuit partitions of a connected 2-in, 2-out digraph D can be determined from the interlacement graph of D with respect to a directed Euler circuit; the same is true of the generating functions for other kinds of circuit partitions. The interlace polynomials of Arratia, Bollobás and Sorkin [J. Combin. Theory Ser. B 92 (2004) 199-233; Combinatorica 24 (2004) 567-584] extend the corresponding functions from interlacement graphs to arbitrary graphs. We introduce a multivariate interlace polynomial that is an analogous extension of a mult…
The interlace polynomials introduced by Arratia, Bollobas and Sorkin extend to invariants of graphs with vertex weights, and these weighted interlace polynomials have several novel properties. One novel property is a version of the fundamental three-term formula q(G)=q(G-a)+q(G^{ab}-b)+((x-1)^{2}-1)q(G^{ab}-a-b) that lacks the last term. It follows that interlace polynomial computations can be represented by binary trees rather than mixed binary-ternary trees. Binary computation trees provide a description of $q(G)$ that is analogous to the activities description of the Tutte polynomial. If $G…
A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality relating the number of circuits in a directed circuit partition of a 2-in, 2-out digraph to the GF(2)-nullity of an associated matrix. This equality is essentially equivalent to the relationship between directed circuit partitions of 2-in, 2-out digraphs and vertex-nullity interlace polynomials of interlace graphs. We present an extension of the Cohn-Lempel equality that describes arbitrary circuit partitions in (undirected) 4-regular graphs. The extended equality incorporates topological results that ha…
We generalize theories of graph, matroid, and ribbon-graph activities to delta-matroids. As a result, we obtain an activities based feasible-set expansion for a transition polynomial of delta-matroids defined by Brijder and Hoogeboom. This result yields feasible-set expansions for the two-variable Bollobás-Riordan and interlace polynomials of a delta-matroid. In the former case, the expansion obtained directly generalizes the activities expansions of the Tutte polynomial of graphs and 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 g…
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.
The interlace polynomial q was introduced by Arratia, Bollobas, and Sorkin. It encodes many properties of the orbit of a graph under edge local complementation (ELC). The interlace polynomial Q, introduced by Aigner and van der Holst, similarly contains information about the orbit of a graph under local complementation (LC). We have previously classified LC and ELC orbits, and now give an enumeration of the corresponding interlace polynomials of all graphs of order up to 12. An enumeration of all circle graphs of order up to 12 is also given. We show that there exist graphs of all orders great…
Multimatroids generalize matroids, delta-matroids, and isotropic systems, and transition polynomials of multimatroids subsume various polynomials for these latter combinatorial structures, such as the interlace polynomial and the Tutte-Martin polynomial. We prove evaluations of the Tutte-Martin polynomial of isotropic systems from Bouchet directly and more efficiently in the context of transition polynomials of multimatroids. Moreover, we generalize some related evaluations of the transition polynomial of 4-regular graphs from Jaeger to multimatroids. These evaluations are obtained in a unifor…
The reconstruction conjecture has remained open for simple undirected graphs since it was suggested in 1941 by Kelly and Ulam. In an attempt to prove the conjecture, many graph invariants have been shown to be reconstructible from the vertex-deleted deck, and in particular, some prominent graph polynomials. Among these are the Tutte polynomial, the chromatic polynomial and the characteristic polynomial. We show that the interlace polynomial, the U -polynomial, the universal edge elimination polynomial xi and the colored versions of the latter two are reconstructible.
In this paper, we prove the real-rootedness of two classes of generalized Narayana polynomials: one arising as the $h$-polynomials of the generalized associahedron associated to the finite Weyl groups, the other arising in the study of the infinite log-concavity of the Boros-Moll polynomials. For the former, Brändén has already proved that these $h$-polynomials have only real zeros. We establish certain recurrence relations for the two classes of Narayana polynomials, from which we derive the real-rootedness. To prove the real-rootedness, we use a sufficient condition, due to Liu and Wang, to …
In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of D. P. Ilyutko an…
Motivated by the `subgraphs world' view of the ferromagnetic Ising model, we develop a general approach to studying mixing times of Glauber dynamics based on subset expansion expressions for a class of graph polynomials. With a canonical paths argument, we demonstrate that the chains defined within this framework mix rapidly upon graphs of bounded tree-width. This extends known results on rapid mixing for the Tutte polynomial, the adjacency-rank ($R_2$-)polynomial and the interlace polynomial.
The Laplacian matching polynomial of a graph $G$, denoted by $\mathscr{L\hspace{-0.7mm}M}(G,x)$, is a new graph polynomial whose all roots are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let $G$ be a connected graph. We show that $0$ is a root of $\mathscr{L\hspace{-0.7mm}M}(G, x)$ if and only if $G$ is a tree. We prove that the number of distinct positive zeros of $\mathscr{L\hspace{-0.7mm}M}(G,x)$ is at least equal to the length of the longest path in $G$. It is also established that the zeros of $\mathscr{L\hspace{-0.7…
In this paper we introduce a family of two-variable derivative polynomials for tangent and secant. We study the generating functions for the coefficients of this family of polynomials. In particular, we establish a connection between these generating functions and Eulerian polynomials.
In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.
