On trees with the same restricted <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="mml9" display="inline" overflow="scroll" altimg="si9.gif"><mml:mi>U</mml:mi></mml:math>-polynomial and the Prouhet–Tarry–Escott problem

Aliste-Prieto, José; Mier, Anna de; Zamora, José

doi:10.1016/j.disc.2016.09.019

Cited by 28 publications

(33 citation statements)

References 15 publications

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“…In particular we prove the relations stated in Theorem 1.2 between the B-polynomial and the Tutte polynomial. We also define two invariants T (1) D (x, y) and T (2) D (x, y) in terms of the specialization B D (q, y, 1), and we explain their relations to the Tutte polynomial of graphs and the chromatic polynomial of mixed graphs. In Section 4, we explore the consequences of recurrence relations for the B-polynomial.…”

Section: Introductionmentioning

confidence: 99%

“…We also prove that the B-polynomial of a planar digraph and its dual satisfy a partial symmetry relation. In Section 7, we interpret several evaluations of the invariants T (1) D (x, y) and T (2) D (x, y) in terms of orientations of mixed graphs. In Section 8, we present a quasisymmetric function generalization of the B-polynomial, and use the theory of P -partitions to prove some new properties.…”

Section: Introductionmentioning

confidence: 99%

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Tutte polynomials for directed graphs

Awan¹,

Bernardi²

2020

Journal of Combinatorial Theory, Series B

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The Tutte polynomial is a fundamental invariant of graphs. In this article, we define and study a generalization of the Tutte polynomial for directed graphs, that we name the B-polynomial. The B-polynomial has three variables, but when specialized to the case of graphs (that is, digraphs where arcs come in pairs with opposite directions), one of the variables becomes redundant and the B-polynomial is equivalent to the Tutte polynomial. We explore various properties, expansions, specializations, and generalizations of the B-polynomial, and try to answer the following questions:• what properties of the digraph can be detected from its B-polynomial (acyclicity, length of directed paths, number of strongly connected components, etc.)? • which of the marvelous properties of the Tutte polynomial carry over to the directed graph setting? The B-polynomial generalizes the strict chromatic polynomial of mixed graphs introduced by Beck, Bogart and Pham. We also consider a quasisymmetric function version of the B-polynomial which simultaneously generalizes the Tutte symmetric function of Stanley and the quasisymmetric chromatic function of Shareshian and Wachs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Tutte polynomials for directed graphs

Awan¹,

Bernardi²

2020

Journal of Combinatorial Theory, Series B

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“…and we may also permute the colors among those L i that have the same cardinality. Since also clearly every proper κ with x κ (G, w) = x λ 1 1 · · · · · x λ k k has a corresponding element of St λ (G, w) for which κ is monochromatic on each part, the terms of X (G,w) are in one-to-one correspondence with those of the right-hand side of (2), so the lemma is proved.…”

Section: Extending X G To Vertex-weighted Graphsmentioning

confidence: 84%

A deletion–contraction relation for the chromatic symmetric function

Crew

Spirkl

2020

European Journal of Combinatorics

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We extend the definition of the chromatic symmetric function X G to include graphs G with a vertex-weight function w : V (G) → N. We show how this provides the chromatic symmetric function with a natural deletion-contraction relation analogous to that of the chromatic polynomial. Using this relation we derive new properties of the chromatic symmetric function, and we give alternate proofs of many fundamental properties of X G .

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“…13]. There are also connections to Ramanujan identities [199,203], other types of multigrade equations [186,205], problems in algebra [198,201], geometry [190], combinatorics [178,179,182,185], graph theory [193], and computer science [183,184,191]. For background information see [4,12,13,18,23].…”

Section: Switching Components and A Problem In Number Theorymentioning

confidence: 99%

13. On the reconstruction of static and dynamic discrete structures

Alpers¹,

Gritzmann²

2019

The Radon Transform

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We study inverse problems of reconstructing static and dynamic discrete structures from tomographic data (with a special focus on the 'classical' task of reconstructing finite point sets in R d ). The main emphasis is on recent mathematical developments and new applications, which emerge in scientific areas such as physics and materials science, but also in inner mathematical fields such as number theory, optimization, and imaging. Along with a concise introduction to the field of discrete tomography, we give pointers to related aspects of computerized tomography in order to contrast the worlds of continuous and discrete inverse problems.

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On trees with the same restricted U-polynomial and the Prouhet–Tarry–Escott problem

Cited by 28 publications

References 15 publications

Tutte polynomials for directed graphs

Tutte polynomials for directed graphs

A deletion–contraction relation for the chromatic symmetric function

13. On the reconstruction of static and dynamic discrete structures

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