Revisiting the Top-Down Computation of BDD of Spanning Trees of a Graph and Its Tutte Polynomial

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 $M$ .

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Top-Down Construction of Independent Spanning Trees in Alternating Group Networks

IEEE Access ◽

10.1109/access.2020.2999421 ◽

2020 ◽

Vol 8 ◽

pp. 112333-112347 ◽

Cited By ~ 2

Author(s):

Jie-Fu Huang ◽

Shih-Shun Kao ◽

Sun-Yuan Hsieh ◽

Ralf Klasing

Keyword(s):

Spanning Trees ◽

Alternating Group ◽

Top Down ◽

Independent Spanning Trees

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The Tutte Polynomial of a Graph, Depth-first Search

The Electronic Journal of Combinatorics ◽

10.37236/1267 ◽

1995 ◽

Vol 3 (2) ◽

Cited By ~ 14

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.

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The Tutte polynomial and toric Nakajima quiver varieties

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2021.61 ◽

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.

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The Computational Complexity of the Tutte Plane: the Bipartite Case

Combinatorics Probability Computing ◽

10.1017/s0963548300000195 ◽

1992 ◽

Vol 1 (2) ◽

pp. 181-187 ◽

Cited By ~ 38

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.

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

The Electronic Journal of Combinatorics ◽

10.37236/833 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 16

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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Top-Down Processing and Visual Reorientation Illusions in a Virtual Reality Environment

Swiss Journal of Psychology ◽

10.1024/1421-0185.63.3.143 ◽

2004 ◽

Vol 63 (3) ◽

pp. 143-149 ◽

Cited By ~ 3

Author(s):

Fred W. Mast ◽

Charles M. Oman

Keyword(s):

Virtual Reality ◽

Visual Processing ◽

Line Length ◽

Perceptual Cues ◽

Virtual Reality Environment ◽

Top Down ◽

Test Conditions ◽

The Subject ◽

Processing Mechanisms

The role of top-down processing on the horizontal-vertical line length illusion was examined by means of an ambiguous room with dual visual verticals. In one of the test conditions, the subjects were cued to one of the two verticals and were instructed to cognitively reassign the apparent vertical to the cued orientation. When they have mentally adjusted their perception, two lines in a plus sign configuration appeared and the subjects had to evaluate which line was longer. The results showed that the line length appeared longer when it was aligned with the direction of the vertical currently perceived by the subject. This study provides a demonstration that top-down processing influences lower level visual processing mechanisms. In another test condition, the subjects had all perceptual cues available and the influence was even stronger.

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Functional Anatomy of Intensity Aspects of Attention

Zeitschrift für Neuropsychologie ◽

10.1024/1016-264x.14.3.181 ◽

2003 ◽

Vol 14 (3) ◽

pp. 181-190 ◽

Cited By ~ 2

Author(s):

Walter Sturm

Keyword(s):

Right Hemisphere ◽

Functional Anatomy ◽

Fmri Data ◽

Functional Networks ◽

Top Down ◽

Stimulus Modality ◽

Spatial Orienting ◽

Phasic Alertness ◽

Co Activation ◽

Attention System

Abstract: Behavioral and PET/fMRI-data are presented to delineate the functional networks subserving alertness, sustained attention, and vigilance as different aspects of attention intensity. The data suggest that a mostly right-hemisphere frontal, parietal, thalamic, and brainstem network plays an important role in the regulation of attention intensity, irrespective of stimulus modality. Under conditions of phasic alertness there is less right frontal activation reflecting a diminished need for top-down regulation with phasic extrinsic stimulation. Furthermore, a high overlap between the functional networks for alerting and spatial orienting of attention is demonstrated. These findings support the hypothesis of a co-activation of the posterior attention system involved in spatial orienting by the anterior alerting network. Possible implications of these findings for the therapy of neglect are proposed.

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Auswirkungen eines Aufmerksamkeitstrainings auf die aphasische Symptomatik bei Schlaganfallpatienten

Zeitschrift für Neuropsychologie ◽

10.1024/1016-264x/a000027 ◽

2011 ◽

Vol 22 (1) ◽

pp. 21-32 ◽

Cited By ~ 2

Author(s):

Julia Graf ◽

Hartwig Kulke ◽

Christa Sous-Kulke ◽

Wilfried Schupp ◽

Stefan Lautenbacher

Keyword(s):

Top Down

Aufmerksamkeit kann als Kontrollsystem neuronaler Aktivität verstanden werden, welches Neuroplastizität top-down modulieren hilft. Bisher wurde selten versucht, durch deren gezielte Förderung Funktionswiederherstellungen nach Hirnschädigung zu begünstigen. In vorliegender Studie wurde dies am Beispiel der Aphasie erprobt. 15 Schlaganfallpatienten erhielten ein dreiwöchiges Training der selektiven Aufmerksamkeit mit den PC-Programmen CogniPlus und „Konzentration“ bei fünf Sitzungen pro Woche zusätzlich zur Standardtherapie, 13 weitere bildeten eine Kontrollgruppe ohne Aufmerksamkeitstraining. Zur Effektivitätskontrolle dienten zwei Versionen des Untertests Go/Nogo (Testbatterie zur Aufmerksamkeitsprüfung) und die Kurze Aphasieprüfung. Nach dem Training manifestierte sich zwischen den Untersuchungsgruppen kein Unterschied in Aufmerksamkeits- und Sprachfunktionen; das zusätzliche Aufmerksamkeitstraining war also wirkungslos. Allerdings zeigten Patienten mit deutlichen Aufmerksamkeitsverbesserungen tendenziell weniger Aphasie-Symptome, was die Hypothese aufmerksamkeitsvermittelter Plastizitätsmodulation nach Hirnschädigung partiell stützt.

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