scholarly journals Generalized bijective maps between G-parking functions, spanning trees, and the Tutte polynomial

2021 ◽  
Author(s):  
Carrie Frizzell
Keyword(s):  
Spanning Trees ◽  
Tutte Polynomial ◽  
Parking Functions

scholarly journals The Tutte Polynomial of a Graph, Depth-first Search

10.37236/1267 ◽  
1995 ◽  
Vol 3 (2) ◽  
Author(s):  
Ira M. Gessel ◽  
Bruce E. Sagan
Keyword(s):  
Spanning Trees ◽  
Tutte Polynomial ◽  
Simplicial Complexes ◽  
Parking Functions ◽  
Depth First Search ◽  
Acyclic Orientations

One of the most important numerical quantities that can be computed from a graph $G$ is the two-variable Tutte polynomial. Specializations of the Tutte polynomial count various objects associated with $G$, e.g., subgraphs, spanning trees, acyclic orientations, inversions and parking functions. We show that by partitioning certain simplicial complexes related to $G$ into intervals, one can provide combinatorial demonstrations of these results. One of the primary tools for providing such a partition is depth-first search.

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 $M$ .

scholarly journals G-parking functions, acyclic orientations and spanning trees

2010 ◽  
Vol 310 (8) ◽  
pp. 1340-1353 ◽  
Author(s):  
Brian Benson ◽  
Deeparnab Chakrabarty ◽  
Prasad Tetali
Keyword(s):  
Spanning Trees ◽  
Parking Functions ◽  
Acyclic Orientations

An integer sequence and standard monomials

2018 ◽  
Vol 17 (02) ◽  
pp. 1850037
Author(s):  
Ajay Kumar ◽  
Chanchal Kumar
Keyword(s):  
Spanning Trees ◽  
Monomial Ideal ◽  
Oriented Graph ◽  
Barycentric Subdivision ◽  
Parking Functions ◽  
Minimal Free Resolution ◽  
Ideal Formula ◽  
Laplace Matrix ◽  
Integer Sequence ◽  
Standard Monomials

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].

scholarly journals The Tutte polynomial and toric Nakajima quiver varieties

2021 ◽  
pp. 1-17
Author(s):  
Tarig Abdelgadir ◽  
Anton Mellit ◽  
Fernando Rodriguez Villegas
Keyword(s):  
Spanning Trees ◽  
Tutte Polynomial ◽  
Cell Decomposition ◽  
Direct Relation ◽  
Poincaré Polynomial ◽  
Quiver Variety ◽  
Quiver Varieties ◽  
Underlying Graph ◽  
Hands On ◽  
A Cell

For a quiver $Q$ with underlying graph $\Gamma$ , we take $ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of $\Gamma$ , the Kac polynomial of $Q$ and the Poincaré polynomial of $ {\mathcal {M}}$ . We do this by giving a cell decomposition of $ {\mathcal {M}}$ indexed by spanning trees of $\Gamma$ and ‘geometrizing’ the deletion and contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] and Crawley-Boevey–Van den Bergh [3], however the methods here are more hands-on.

The Computational Complexity of the Tutte Plane: the Bipartite Case

1992 ◽  
Vol 1 (2) ◽  
pp. 181-187 ◽  
Author(s):  
D. L. Vertigan ◽  
D. J. A. Welsh
Keyword(s):  
Computational Complexity ◽  
Ising Model ◽  
Partition Function ◽  
Polynomial Time ◽  
Spanning Trees ◽  
Tutte Polynomial ◽  
Wide Range ◽  
Number Of Spanning Trees ◽  
Bipartite Case ◽  
Complete Characterisation

Along different curves and at different points of the (x, y)-plane the Tutte polynomial evaluates a wide range of quantities. Some of these, such as the number of spanning trees of a graph and the partition function of the planar Ising model, can be computed in polynomial time, others are # P-hard. Here we give a complete characterisation of which points and curves are easy/hard in the bipartite case.

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.

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