Evaluations of Graph Polynomials

Motivated by the definition of the edge elimination polynomial of a graph we define the covered components polynomial counting spanning subgraphs with respect to their number of components, edges and covered components. We prove a recurrence relation, which shows that both graph polynomials are substitution instances of each other. We give some properties of the covered components polynomial and some results concerning relations to other graph polynomials.

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Graph polynomials and symmetries

Journal of Algebra and Its Applications ◽

10.1142/s021949881950172x ◽

2019 ◽

Vol 18 (09) ◽

pp. 1950172 ◽

Cited By ~ 1

Author(s):

Nafaa Chbili

Keyword(s):

Automorphism Group ◽

Finite Field ◽

Characteristic Polynomial ◽

Prime Order ◽

Tutte Polynomial ◽

Quotient Graph ◽

Graph Polynomials ◽

Tutte Polynomials

In a recent paper, we studied the interaction between the automorphism group of a graph and its Tutte polynomial. More precisely, we proved that certain symmetries of graphs are clearly reflected by their Tutte polynomials. The purpose of this paper is to extend this study to other graph polynomials. In particular, we prove that if a graph [Formula: see text] has a symmetry of prime order [Formula: see text], then its characteristic polynomial, with coefficients in the finite field [Formula: see text], is determined by the characteristic polynomial of its quotient graph [Formula: see text]. Similar results are also proved for some generalization of the Tutte polynomial.

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Graphs, ribbon graphs, and polynomials

Combinatorial Physics ◽

10.1093/oso/9780192895493.003.0002 ◽

2021 ◽

pp. 7-16

Author(s):

Adrian Tanasa

Keyword(s):

Graph Theory ◽

Tutte Polynomial ◽

Graph Polynomials ◽

Ribbon Graphs

In this chapter we present some notions of graph theory that will be useful in the rest of the book. It is worth emphasizing that graph theorists and theoretical physicists adopt, unfortunately, different terminologies. We present here both terminologies, such that a sort of dictionary between these two communities can be established. We then extend the notion of graph to that of maps (or of ribbon graphs). Moreover, graph polynomials encoding these structures (the Tutte polynomial for graphs and the Bollobás–Riordan polynomial for ribbon graphs) are presented.

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Thermodynamic characterization of networks using graph polynomials

Physical Review E ◽

10.1103/physreve.92.032810 ◽

2015 ◽

Vol 92 (3) ◽

Cited By ~ 15

Author(s):

Cheng Ye ◽

César H. Comin ◽

Thomas K. DM. Peron ◽

Filipi N. Silva ◽

Francisco A. Rodrigues ◽

...

Keyword(s):

Thermodynamic Characterization ◽

Graph Polynomials

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Semantic Equivalence of Graph Polynomials Definable in Second Order Logic

Logic, Language, Information, and Computation - Lecture Notes in Computer Science ◽

10.1007/978-3-662-52921-8_18 ◽

2016 ◽

pp. 279-296

Author(s):

Johann A. Makowsky ◽

Elena V. Ravve

Keyword(s):

Second Order ◽

Order Logic ◽

Semantic Equivalence ◽

Graph Polynomials ◽

Second Order Logic

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Graph polynomials and link invariants as positive type functions on Thompson’s group F

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216519500068 ◽

2019 ◽

Vol 28 (02) ◽

pp. 1950006 ◽

Cited By ~ 2

Author(s):

Valeriano Aiello ◽

Roberto Conti

Keyword(s):

Elementary Proof ◽

Tutte Polynomial ◽

Positive Type ◽

Polynomial Invariants ◽

Graph Polynomials ◽

Link Invariants ◽

Kauffman Bracket ◽

Model Interpretation ◽

Mechanics Model ◽

Thompson’S Group

In a recent paper, Jones introduced a correspondence between elements of the Thompson group [Formula: see text] and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be reinterpreted as coefficients of certain unitary representations of [Formula: see text]. We give a somewhat different and elementary proof of this fact for the Kauffman bracket evaluated at certain roots of unity by means of a statistical mechanics model interpretation. Moreover, by similar methods we show that, for some particular specializations of the variables, other familiar link invariants and graph polynomials, namely the number of [Formula: see text]-colorings and the Tutte polynomial, can be viewed as positive definite functions on [Formula: see text].

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