scholarly journals Tutte Polynomial, Subgraphs, Orientations and Sandpile Model: New Connections via Embeddings

10.37236/833 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Olivier Bernardi
Keyword(s):  
Spanning Trees ◽  
Tutte Polynomial ◽  
Cyclic Order ◽  
Sandpile Model ◽  
Spanning Subgraphs ◽  
Connected Subgraphs

We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations $T_G(i,j),0\leq i,j \leq 2$ of the Tutte polynomial in terms of subgraphs, orientations, outdegree sequences and sandpile configurations. For instance, for any graph $G$, we obtain a bijection between connected subgraphs (counted by $T_G(1,2)$) and root-connected orientations, a bijection between forests (counted by $T_G(2,1)$) and outdegree sequences and bijections between spanning trees (counted by $T_G(1,1)$), root-connected outdegree sequences and recurrent sandpile configurations. All our proofs are based on a single bijection $\Phi$ between the spanning subgraphs and the orientations that we specialize in various ways. The bijection $\Phi$ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex.

scholarly journals On the characterization of graphs with maximum number of spanning trees

1998 ◽  
Vol 179 (1-3) ◽  
pp. 155-166 ◽  
Author(s):  
L. Petingi ◽  
F. Boesch ◽  
C. Suffel
Keyword(s):  
Spanning Trees ◽  
Number Of Spanning Trees

scholarly journals Analysis of Markov chain algorithms on spanning trees, rooted forests, and connected subgraphs

2005 ◽  
Vol 32 (3) ◽  
pp. 341-365 ◽  
Author(s):  
Johannes Fehrenbach ◽  
Ludger Rüschendorf
Keyword(s):  
Markov Chain ◽  
Spanning Trees ◽  
Connected Subgraphs

scholarly journals GEOMETRIC BIJECTIONS FOR REGULAR MATROIDS, ZONOTOPES, AND EHRHART THEORY

2019 ◽  
Vol 7 ◽  
Author(s):  
SPENCER BACKMAN ◽  
MATTHEW BAKER ◽  
CHI HO YUEN
Keyword(s):  
Spanning Trees ◽  
Finite Abelian Group ◽  
Tutte Polynomial ◽  
Geometric Interpretation ◽  
Lattice Points ◽  
Combinatorial Interpretation ◽  
Ehrhart Polynomial ◽  
Ehrhart Theory ◽  
Regular Matroid ◽  
Definition Of

Let $M$ be a regular matroid. The Jacobian group $\text{Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$ . This group generalizes the definition of the Jacobian group (also known as the critical group or sandpile group) $\operatorname{Jac}(G)$ of a graph $G$ (in which case bases of the corresponding regular matroid are spanning trees of $G$ ). There are many explicit combinatorial bijections in the literature between the Jacobian group of a graph $\text{Jac}(G)$ and spanning trees. However, most of the known bijections use vertices of $G$ in some essential way and are inherently ‘nonmatroidal’. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid $M$ and bases of $M$ , many instances of which are new even in the case of graphs. We first describe our family of bijections in a purely combinatorial way in terms of orientations; more specifically, we prove that the Jacobian group of $M$ admits a canonical simply transitive action on the set ${\mathcal{G}}(M)$ of circuit–cocircuit reversal classes of $M$ , and then define a family of combinatorial bijections $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ between ${\mathcal{G}}(M)$ and bases of $M$ . (Here $\unicode[STIX]{x1D70E}$ (respectively $\unicode[STIX]{x1D70E}^{\ast }$ ) is an acyclic signature of the set of circuits (respectively cocircuits) of $M$ .) We then give a geometric interpretation of each such map $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70E}^{\ast }}$ in terms of zonotopal subdivisions which is used to verify that $\unicode[STIX]{x1D6FD}$ is indeed a bijection. Finally, we give a combinatorial interpretation of lattice points in the zonotope $Z$ ; by passing to dilations we obtain a new derivation of Stanley’s formula linking the Ehrhart polynomial of $Z$ to the Tutte polynomial of $M$ .

Decomposing hypercubes into regular connected subgraphs

2016 ◽  
Vol 08 (04) ◽  
pp. 1650065 ◽  
Author(s):  
A. V. Sonawane ◽  
Y. M. Borse
Keyword(s):  
Spanning Subgraphs ◽  
Connected Subgraphs

It is known that the [Formula: see text]-dimensional hypercube [Formula: see text] for [Formula: see text] with [Formula: see text] can be decomposed into two spanning bipancyclic subgraphs [Formula: see text] and [Formula: see text] such that [Formula: see text] is [Formula: see text]-regular and [Formula: see text]-connected for [Formula: see text] In this paper, we prove that if [Formula: see text] with [Formula: see text] and at most one [Formula: see text] odd, then [Formula: see text] can be decomposed into [Formula: see text] spanning subgraphs [Formula: see text], [Formula: see text] such that [Formula: see text] is [Formula: see text]-regular and [Formula: see text]-connected for [Formula: see text]

scholarly journals Permutation Graphs and the Abelian Sandpile Model, Tiered Trees and Non-Ambiguous Binary Trees

10.37236/8225 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Mark Dukes ◽  
Thomas Selig ◽  
Jason P. Smith ◽  
Einar Steingrímsson
Keyword(s):  
Tutte Polynomial ◽  
Binary Trees ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Sandpile Model ◽  
Abelian Sandpile Model ◽  
Threshold Graphs ◽  
Famous Result ◽  
Abelian Sandpile ◽  
Ferrers Graphs

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 result linking the level polynomial of the ASM to the ubiquitous Tutte polynomial. We show that the set of minimal recurrent configurations is in bijection with the set of complete non-ambiguous binary trees introduced by Aval et al. [2], and introduce a multi-rooted generalization of these that we show to correspond to all recurrent configurations. In the case of permutations with a single descent, we recover some results from the case of Ferrers graphs presented in [11], while we also recover results of Perkinson et al. [16] in the case of threshold graphs.

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