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Stable multivariate generalizations of matching polynomials
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).ORCID iD: 0000-0002-2305-9764
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan d-coverings, Hall, Puder and Sawin introduced the d-matching polynomial of a graph G, defined as the uniform average of matching polynomials over the set of d-sheeted covering graphs of G. We prove that a natural multivariate version of the d-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the d-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.

Keywords [en]
generalized matching polynomial, stable polynomial, hypergraph
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-250765OAI: oai:DiVA.org:kth-250765DiVA, id: diva2:1313632
Note

QC 20190510

Available from: 2019-05-05 Created: 2019-05-05 Last updated: 2019-05-13Bibliographically approved
In thesis
1. Combinatorics and zeros of multivariate polynomials
Open this publication in new window or tab >>Combinatorics and zeros of multivariate polynomials
2019 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of five papers in algebraic and enumerative combinatorics. The objects at the heart of the thesis are combinatorial polynomials in one or more variables. We study their zeros, coefficients and special evaluations. Hyperbolic polynomials may be viewed as multivariate generalizations of real-rooted polynomials in one variable. To each hyperbolic polynomial one may associate a convex cone from which a matroid can be derived - a so called hyperbolic matroid. In Paper A we prove the existence of an infinite family of non-representable hyperbolic matroids parametrized by hypergraphs. We further use special members of our family to investigate consequences to a central conjecture around hyperbolic polynomials, namely the generalized Lax conjecture. Along the way we strengthen and generalize several symmetric function inequalities in the literature, such as the Laguerre-Tur\'an inequality and an inequality due to Jensen. In Paper B we affirm the generalized Lax conjecture for two related classes of combinatorial polynomials: multivariate matching polynomials over arbitrary graphs and multivariate independence polynomials over simplicial graphs. In Paper C we prove that the multivariate $d$-matching polynomial is hyperbolic for arbitrary multigraphs, in particular answering a question by Hall, Puder and Sawin. We also provide a hypergraphic generalization of a classical theorem by Heilmann and Lieb regarding the real-rootedness of the matching polynomial of a graph. In Paper D we establish a number of equidistributions between Mahonian statistics which are given by conic combinations of vincular pattern functions of length at most three, over permutations avoiding a single classical pattern of length three. In Paper E we find necessary and sufficient conditions for a candidate polynomial to be complemented to a cyclic sieving phenomenon (without regards to combinatorial context). We further take a geometric perspective on the phenomenon by associating a convex rational polyhedral cone which has integer lattice points in correspondence with cyclic sieving phenomena. We find the half-space description of this cone and investigate its properties.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2019. p. 42
Series
TRITA-SCI-FOU ; 2019:33
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-251303 (URN)978-91-7873-210-4 (ISBN)
Public defence
2019-05-24, D3, Lindstedsvägen 5, Stockholm, 14:00 (English)
Opponent
Supervisors
Note

QC 20190510

Available from: 2019-05-10 Created: 2019-05-09 Last updated: 2019-05-10Bibliographically approved

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Amini, Nima

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