Tutte Polynomials for Directed Graphs

Awan, Jordan; Bernardi, Olivier

doi:10.48550/arxiv.1610.01839

Cited by 3 publications

(12 citation statements)

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“…, where π g (n) is the weak-chromatic polynomial defined in [3] and I is the vertex set of g. This fact is already established in [3], but our proof puts it in a new context. In [3], they also define a 3-variable polynomial invariant B g (n, x, y) for directed graphs g. We call this the B-polynomial. The strict-and weak-chromatic polynomials are specializations of the B-polynomial.…”

Section: Introductionsupporting

confidence: 60%

“…The presence of such would not change any of our constructions. We will denote by g = (I, E) the directed graph g with vertex set I and directed edge set E. Definition 2.35 (Chromatic polynomial of [3]). Let g = (I, E) ∈ DG[I] be a directed graph where E is the directed edge set of g. The strict-chromatic polynomial…”

Section: 4mentioning

confidence: 99%

“…Furthermore, P(z g ) provides a morphism from the Hopf monoid of directed graphs to the Hopf monoid of generalized permutahedra. From Theorem 1.1, we derive that the AA polynomial obtained from the basic character of directed graphs is equivalent to the strict-chromatic polynomial π > g (n) [3]. Theorem 1.2 (main theorem).…”

Section: Introductionmentioning

confidence: 99%

“…. The strict-chromatic polynomial is defined by Awan and Bernardi [3] to study properties of directed graphs (see Section 2.6). From the reciprocity theorem of Hopf monoids, it follows that we have (−1…”

Section: Introductionmentioning

confidence: 99%

“…In subsections 2.1-2.5, we introduce further definitions and some general facts including Aguiar-Ardila's results in [1]. In subsection 2.6, we recall Awan-Bernardi's results in [3]. In section 3, we introduce the Hopf monoid of directed graphs and we prove Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The Hopf monoid and the basic invariant of directed graphs

Kato

2019

Preprint

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Aguiar and Ardila defined the Hopf monoid GP of generalized permutahedra and showed that it contains many submonoids that correspond to combinatorial objects. They also give a basic polynomial invariant of generalized permutahedra, which then specializes to the submonoids. We define the Hopf monoid of directed graphs and show that it also embeds in GP. The resulting basic invariant coincides with the strict chromatic polynomial of Awan and Bernardi.

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Section: Introductionsupporting

confidence: 60%

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The Hopf monoid and the basic invariant of directed graphs

Kato

2019

Preprint

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Higher matrix-tree theorems and Bernardi polynomial

Burman

2019

J Algebr Comb

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This paper is a continuation of [2]. We prove a three-parameter family of identities (Theorem 1.1) involving a version of the Tutte polynomial for directed graphs introduced by Awan and Bernardi [1]. A particular case of this family (Corollary 1.6) is the higher-degree generalization of the matrixtree theorem proved in [2], which thus receives a new proof, shorter (and less direct) than the original one. The theory has a parallel version for undirected graphs (Theorem 1.2).

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$P$-partitions and $p$-positivity

Alexandersson,

Sulzgruber

2018

Preprint

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Using the combinatorics of α-unimodal sets, we establish two new results in the theory of quasisymmetric functions. First, we obtain the expansion of the fundamental basis into quasisymmetric power sums. Secondly, we prove that generating functions of reverse P -partitions expand positively into quasisymmetric power sums. Consequently any nonnegative linear combination of such functions is p-positive whenever it is symmetric. As an application we derive positivity results for chromatic quasisymmetric functions, unicellular and vertical strip LLT polynomials, multivariate Tutte polynomials and the more general B-polynomials, matroid quasisymmetric functions, and certain Eulerian quasisymmetric functions, thus reproving and improving on numerous results in the literature.

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Tutte Polynomials for Directed Graphs

Cited by 3 publications

References 0 publications

The Hopf monoid and the basic invariant of directed graphs

The Hopf monoid and the basic invariant of directed graphs

Higher matrix-tree theorems and Bernardi polynomial

$P$-partitions and $p$-positivity

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