The Covered Components Polynomial: A New Representation of the Edge Elimination Polynomial

Trinks, Martin

doi:10.37236/2072

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…Straightforwardly generalizing the de nitions and replacing "graph" by "hypergraph", all the results stating recurrence relations and the encoding of the degree sequence, that is all results except Theorem 8 and Theorem 10, stay valid (including the proofs). Results concerning the edge elimination polynomial of hypergraphs are also given by White [25] and the present author [19,Section 9].…”

Section: Introductionmentioning

confidence: 88%

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Proving properties of the edge elimination polynomial using equivalent graph polynomials

Trinks¹

2012

Preprint

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Averbouch, Godlin, and Makowsky de ne the edge elimination polynomial of a graph by a recurrence relation with respect to the deletion, contraction and extraction of an edge. It generalizes some well-known graph polynomials such as the chromatic polynomial and the matching polynomial. By introducing two equivalent graph polynomials, one enumerating subgraphs and the other enumerating colorings, we show that the edge elimination polynomial of a simple graph is reconstructible from its polynomial deck and that it encodes the degree sequence of an arbitrary graph.

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Section: Introductionmentioning

confidence: 88%

“…Other graph polynomials equivalent to the edge elimination polynomial are given by White [25], by Averbouch, Kotek, Makowsky, and Ravve [3], and by the present author [19]. Actually, there is one more equivalent graph polynomial in a publication several years ago, the subgraph enumerating polynomial de ned by Borzacchini and Pulito [6].…”

Section: Introductionmentioning

confidence: 98%

Proving properties of the edge elimination polynomial using equivalent graph polynomials

Trinks¹

2012

Preprint

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“…(For a recurrence relation of (−1) n(G) · Ψ(G, −x), see [2,Equation (9)]. ) Equation (33) has already been derived in the case of graphs via other graph polynomials [11,Corollary 30].…”

Section: From This Proposition the Relations Between Both Graph Polynmentioning

confidence: 99%

A Survey on Recurrence Relations for the Independence Polynomial of Hypergraphs

Trinks

2016

Graphs and Combinatorics

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The independence polynomial of a hypergraph is the generating function for its independent (vertex) sets with respect to their cardinality. This article aims to discuss several recurrence relations for the independence polynomial using some vertex and edge operations. Further, an extension of the well-known recurrence relation for simple graphs to hypergraphs is proven and other novel recurrence relations are also discussed.

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“…There are several well-known graph polynomials, e.g. the Tutte polynomial [15,6,3], the matching polynomial [8,5,18], the domination polynomial [1,11] and the edge elimination polynomial [2,14].…”

Section: Introductionmentioning

confidence: 99%

Note on the Subgraph Component Polynomial

Liao

Hou

2014

Electron. J. Combin.

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Tittmann, Averbouch and Makowsky [P. Tittmann, I. Averbouch, J.A. Makowsky, The enumeration of vertex induced subgraphs with respect to the number of components, European Journal of Combinatorics, 32 (2011) 954-974], introduced the subgraph component polynomial Q(G; x, y) which counts the number of connected components in vertex induced subgraphs. It contains much of the underlying graph's structural information, such as the order, the size, the independence number. We show that there are several other graph invariants, like the connectivity, the number of cycles of length four in a regular bipartite graph, are determined by the subgraph component polynomial. We prove that several well-known families of graphs are determined by the polynomial Q(G; x, y). Moreover, we study the distinguishing power and find some simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the character polynomial, the matching polynomial or the Tutte polynomial. These are answers to three problems proposed by Tittmann

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The Covered Components Polynomial: A New Representation of the Edge Elimination Polynomial

Cited by 7 publications

References 22 publications

Proving properties of the edge elimination polynomial using equivalent graph polynomials

Proving properties of the edge elimination polynomial using equivalent graph polynomials

A Survey on Recurrence Relations for the Independence Polynomial of Hypergraphs

Note on the Subgraph Component Polynomial

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