A Note on Graph Colorings and 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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Evaluations of Graph Polynomials

Graph-Theoretic Concepts in Computer Science - Lecture Notes in Computer Science ◽

10.1007/978-3-540-92248-3_17 ◽

2008 ◽

pp. 183-194 ◽

Cited By ~ 7

Author(s):

Benny Godlin ◽

Tomer Kotek ◽

Johann A. Makowsky

Keyword(s):

Graph Polynomials

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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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