Related papers: A version of Tutte's polynomial for hypergraphs
We prove some variants of the exponential formula and apply them to the multivariate Tutte polynomials (also known as Potts-model partition functions) of graphs. We also prove some further identities for the multivariate Tutte polynomial,…
We give a fine-grained classification of evaluating the Tutte polynomial $T(G;x,y)$ on all integer points on graphs with small treewidth and cutwidth. Specifically, we show for any point $(x,y) \in \mathbb{Z}^2$ that either - can be…
A fourientation of a graph is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. Fixing a total order on the edges and a reference orientation of the graph, we…
We introduce the notions of arithmetic colorings and arithmetic flows over a graph with labelled edges, which generalize the notions of colorings and flows over a graph. We show that the corresponding arithmetic chromatic polynomial and…
We provide a matrix-based formula for the Tutte symmetric function of a graph. In particular, for any graph $G$ with a designated head and tail vertex, we describe an infinite matrix $M_G$ from which the Tutte symmetric function can be…
An oriented hypergraph is an oriented incidence structure that allows for the generalization of graph theoretic concepts to integer matrices through its locally signed graphic substructure. The locally graphic behaviors are formalized in…
Stanley's symmetrized chromatic polynomial is a generalization of the ordinary chromatic polynomial to a graph invariant with values in a ring of polynomials in infnitely many variables. The ordinary chromatic polynomial is a specialization…
We construct a new polynomial invariant of maps (graphs embedded in a compact surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollob\'as--Riordan polynomial, the Las Vergnas polynomial,…
We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem…
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have…
We look for graph polynomials which satisfy recurrence relations on three kinds of edge elimination: edge deletion, edge contraction and deletion of edges together with their end points. Like in the case of deletion and contraction only (W.…
In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be…
Employing two models, we show that various counting functions of a random variable defined by restriction or contraction of a ranked set with multiplicity (e.g., classical and arithmetic matroids) have expectations given by the…
Many graph polynomials, such as the Tutte polynomial, the interlace polynomial and the matching polynomial, have both a recursive definition and a defining subset expansion formula. In this paper we present a general, logic-based framework…
We prove that the Tutte polynomial of a coloopless paving matroid is convex along the portions of the line segments x+y=p lying in the positive quadrant. Every coloopless paving matroids is in the class of matroids which contain two…
Let $G$ be a connected graph with vertex set $\{0,1,2,...,n\}$. We allow $G$ to have multiple edges and loops. In this paper, we give a characterization of external activity by some parameters of $G$-parking functions. In particular, we…
By considering Tutte polynomials of Hopf algebras, we show how a Tutte polynomial can be canonically associated with combinatorial objects that have some notions of deletion and contraction. We show that several graph polynomials from the…
We present those properties of planar doodles, especially when regarded as 4-valent graphs, that enable us to classify them into {\it prime} and {\it super prime} doodles by analogy to a knot sum. We describe a method for partially…
Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic…
In this paper, we give a new proof on a Theorem of Tutte which says that the determinants of the adjacency matrices of two hypomorphic graphs are the same. We also study the lowest eigenvectors.