On the dependence polynomial of a graph

Authors:
Jianguo Qian;Andreas Dress;Yan Wang
Affiliations:
Department of Mathematics, Xiamen University, Xiamen, Fujian, PR China and Department of Mathematics, University of Bielefeld, Bielefeld, Germany;Department of Mathematics, University of Bielefeld, Bielefeld, Germany and Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany;Department of Mathematics, Xiamen University, Xiamen, Fujian, PR China
Venue:
European Journal of Combinatorics
Year:
2007

Citing 4
Cited 1

Dependence polynomials

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Clique polynomials and independent set polynomials of graphs

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Clique polynomials have a unique root of smallest modulus

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Matching Theory (North-Holland mathematics studies)

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Mehler formulae for matching polynomials of graphs and independence polynomials of clawfree graphs

Journal of Combinatorial Theory Series B

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Abstract

The dependence polynomial PG = PG(z) of a graph G is defined by PG(z) := Σi=0n (-1)icizi where ci = ci(G) is the number of complete subgraphs of G of cardinality i. It is clear that the complete subgraphs of G form a poset relative to subset inclusion. Using Möbius inversion, this yields various identities involving dependence polynomials implying in particular that the dependence polynomial of the line graph L(G) of G is determined uniquely by the (multiset of) vertex degrees of G and the number of triangles in G. Furthermore, the dependence polynomial of the complement of the line graph of G is closely related to the matching polynomial of G, one of the most 'prominent' polynomials studied in graph theory.