For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation T_G(2,2)=2^n of its Tutte polynomial. We define a bijection $Φ$ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection $Φ$ is closely related to a recent characterization of the Tutte polynomial relying on a combinatorial embedding of the graph G, that is, on a choice of cyclic order of the edges around each vertex. Among other results, we obtain a combinatorial interpretation for each of the evaluations T_G(i,j), 0\leq i,j \leq 2 of the Tutte polynomial in terms of orientations. The strength of our approach is to derive all these interpretations by specializing the bijection $Φ$ in various ways. For instance, we obtain a bijection between the connected subgraphs of G (counted by T_G(1,2)) and the root-connected orientations. We also obtain a bijection between the forests (counted by T_G(2,1)) and outdegree sequences which specializes into a bijection between spanning trees (counted by T_G(1,1)) and root-connected outdegree sequences. We also define a bijection between spanning trees and recurrent configurations of the sandpile model. Combining our results we obtain a bijection between recurrent configurations and root-connected outdegree sequences which leaves the configurations at level 0 unchanged.
Identities obtained by elementary finite Fourier analysis are used to derive a variety of evaluations of the Tutte polynomial of a graph G at certain points (a,b) where (a-1)(b-1) equals 2 or 4. These evaluations are expressed in terms of eulerian subgraphs of G and the size of subgraphs modulo 2,3,4 or 6. In particular, a graph is found to have a nowhere-zero 4-flow if and only if there is a correlation between the event that three subgraphs A,B,C chosen uniformly at random have pairwise eulerian symmetric differences and the event that the integer part of (|A| + |B| + |C|) / 3 is even. Some …
A fourientation of a graph $G$ is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of $G$ which allow for a new 12 variable expansion of the Tutte polynomial $T_G$. …
Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar graph such that if $G$ has a flat embedding, every possible conflict graph associated to every maximally planar subgraph of $G$ is balanced. In doing this, we show that for every graph $G$ with flat embedding, and a planar subgraph $P$ of $G$, $P$ lies on a sphere that intersects $G$ only in $P$. We conjecture that $G$ is intrinsically linked if and only if eve…
In this paper, we find recursive formulas for the Tutte polynomial of a family of small-world networks: Farey graphs, which are modular and have an exponential degree hierarchy. Then, making use of these formulas, we determine the number of spanning trees, as well as the number of connected spanning subgraphs. Furthermore, we also derive exact expressions for the chromatic polynomial and the reliability polynomial of these graphs.
We introduce a new model of a stochastic sandpile on a graph $G$ containing a sink. When unstable, a site sends one grain to each of its neighbours independently with probability $p \in (0,1]$. For $p=1$, this coincides with the standard Abelian sandpile model. In general, for $p\in(0,1)$, the set of recurrent configurations of this sandpile model is different from that of the Abelian sandpile model. We give a characterisation of this set in terms of orientations of the graph $G$. We also define the lacking polynomial $L_G$ as the generating function counting this set according to the number o…
A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the tiered trees introduced by Dugan et al. [10]. This bijection allows certain parameters of the recurrent configurations to be read on the corresponding tree. In particular, we show that the level of a recurrent configuration can be interpreted as the external activity of the corresponding tree, so that the bijection exhibited provides a new proof of a famous …
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to the hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and …
We study combinatorial aspects of the sandpile model on wheel and fan graphs, seeking bijective characterisations of the model's recurrent configurations on these families. For wheel graphs, we exhibit a bijection between these recurrent configurations and the set of subgraphs of the cycle graph which maps the level of the configuration to the number of edges of the subgraph. This bijection relies on two key ingredients. The first consists in considering a stochastic variant of the standard Abelian sandpile model (ASM), rather than the ASM itself. The second ingredient is a mapping from a give…
Oriented ribbon graphs (dessins d'enfant) are graphs embedded in oriented surfaces. The Bollobás-Riordan-Tutte polynomial is a three-variable polynomial that extends the Tutte polynomial to oriented ribbon graphs. A quasi-tree of a ribbon graph is a spanning subgraph with one face, which is described by an ordered chord diagram. We generalize the spanning tree expansion of the Tutte polynomial to a quasi-tree expansion of the Bollobás-Riordan-Tutte polynomial.
A parking function of length n is a sequence (b_1, b_2,..., b_n) of nonnegative integers whose nondecreasing rearrangement (a_1, a_2,...,a_n) has the property that a_i < i for every i. A well-known result about parking functions is that the polynomial P_n(q), which enumerates the complements of parking functions by the sum of their terms, is the generating function for the number of connected graphs by the number of excess edges when evaluated at (1+q). In this paper we extend this result to arbitrary connected graphs G. In general the polynomial that encodes information about subgraphs of G i…
This thesis deals with the Tutte polynomial, studied from different points of view. In the first part, we address the enumeration of planar maps equipped with a spanning forest, here called forested maps, with a weight $z$ per face and a weight $u$ per non-root component of the forest. Equivalently, we count (with respect to the number of faces) the planar maps $C$ weighted by $T_C(u+1,1)$, where $T_C$ is the Tutte polynomial of $C$. We begin by a purely combinatorial characterization of the corresponding generating function, denoted by $F(z,u)$. We deduce from this that $F(z,u)$ is differenti…
Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial $Q(G;x,y)$ which counts the number of connected components in vertex induced subgraphs. It contains much of the underlying graph's structural information, e.g. the order, the size, the independence number. We show that there are several other graph invariants, like the connectivity, the number of cycles of length four in a regular bipar…
What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G …
Inspired by the study of community structure in connection networks, we introduce the graph polynomial $Q(G;x,y)$, the bivariate generating function which counts the number of connected components in induced subgraphs. We give a recursive definition of $Q(G;x,y)$ using vertex deletion, vertex contraction and deletion of a vertex together with its neighborhood and prove a universality property. We relate $Q(G;x,y)$ to other known graph invariants and graph polynomials, among them partition functions, the Tutte polynomial, the independence and matching polynomials, and the universal edge elimina…
Given a graph G, an incidence matrix N(G) is defined for the set of distinct isomorphism types of induced subgraphs of G. If Ulam's conjecture is true, then every graph invariant must be reconstructible from this matrix, even when the graphs indexing the rows and the columns of N(G) are unspecified. It is proved that the characteristic polynomial, the rank polynomial, and the number of spanning trees of a graph are reconstructible from its N-matrix. These results are stronger than the original results of Tutte in the sense that actual subgraphs are not used. It is also proved that the characte…
Princeton University（普林斯顿大学）；Tel Aviv University（特拉维夫大学）；University of Waterloo（滑铁卢大学）；ETH, Zürich（苏黎世联邦理工学院）；University of California, San Diego（加州大学圣地亚哥分校）
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.
Given a bipartite graph $G$, the graphical matrix space $\mathcal{S}_G$ consists of matrices whose non-zero entries can only be at those positions corresponding to edges in $G$. Tutte (J. London Math. Soc., 1947), Edmonds (J. Res. Nat. Bur. Standards Sect. B, 1967) and Lovász (FCT, 1979) observed connections between perfect matchings in $G$ and full-rank matrices in $\mathcal{S}_G$. Dieudonné ({Arch. Math., 1948) proved a tight upper bound on the dimensions of those matrix spaces containing only singular matrices. The starting point of this paper is a simultaneous generalization of these two c…
