A Survey on Recurrence Relations for the Independence Polynomial of Hypergraphs

For a graph G = (V, E), k ∈ N, and a complex number w the partition function of the univariate Potts model is defined aswhere [k] := {1, . . . , k}. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any ∆ ∈ N and any k ≥ e∆ + 1, there exists an open set U in the complex plane that contains the interval [0, 1) such that Z(G; k, w) = 0 for any w ∈ U and any graph G of maximum degree at most ∆. (Here e denotes the base of the natural logarithm.) For small values of ∆ we are able to give better results.As an application of our results we obtain improved bounds on k for the existence of deterministic approximation algorithms for counting the number of proper k-colourings of graphs of small maximum degree.

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“…Now let us collect basic relations following from the definition of independent sets. We refer to [Tri16] for their proofs.…”

Section: Independence Polynomial Of Hypergraphsmentioning

confidence: 99%

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

Bencs

Davies

Patel

et al. 2021

Ann. Inst. Henri Poincaré Comb. Phys. Interact.

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“…Below we consider only some (not all) questions concerned with clique-type polynomials. Note the recent works devoted to independence and matching polynomials defined for hypergraphs [14,104,149,189,192].…”

Section: Survey Open Problemsmentioning

confidence: 99%

$PC$-polynomial of graph

Gubarev

2018

Preprint

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We define P C-polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root β(G) of P C-polynomial of the corresponding graph.The random algebra is defined in such way that its growth rate equals the largest root of P C-polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of P C-polynomial lie in neighbourhoods of roots of P Cpolynomial of random graph. We show how to calculate the series expansions of the latter roots. The average value of β(G) over all graphs with the same number of vertices is computed.We found the graphs on which the maximal value of β(G) with fixed numbers of vertices and edges is reached. From this, we derive the upper bound of β(G). Modulo one assumption, we do the same for minimal value of β(G).We study the Nordhaus Gaddum bounds of β(G) + β( Ḡ) and β(G)β( Ḡ).

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“…where a k is the number of independence sets of G with exactly k vertices. For more details on the independence polynomial, we refer to [14,16,17,19].…”

Section: The Fundamental Theorem For Matching Polynomialsmentioning

confidence: 99%

On the matching polynomial of hypergraphs

Guo¹,

Zhao²,

Mao³

2017

JACODESMATH

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The concept of the matching polynomial of a graph, introduced by Farrell in 1979, has received considerable attention and research. In this paper, we generalize this concept and introduce the matching polynomial of hypergraphs. A recurrence relation of the matching polynomial of a hypergraph is obtained. The exact matching polynomials of some special hypergraphs are given. Further, we discuss the zeros of matching polynomials of hypergraphs.2010 MSC: 05C31, 05C65, 05C70

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A Survey on Recurrence Relations for the Independence Polynomial of Hypergraphs

Cited by 6 publications

References 7 publications

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs

$PC$-polynomial of graph

On the matching polynomial of hypergraphs

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