The Path-Cycle Symmetric Function of a Digraph

Chow, Timothy Y.

doi:10.1006/aima.1996.0018

Cited by 25 publications

(37 citation statements)

References 20 publications

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“…10. Moreover, V= [1,3] is independent with respect to this ordering, with V < empty and V > = [2], so Theorem 4.1 applies. The boards of D"V and D | V and the factors they determine are shown in Fig.…”

Section: Partial Factorization Of the Cover Polynomial Of Skyline Boardsmentioning

confidence: 82%

Factorization of the Cover Polynomial

Dworkin

1997

Journal of Combinatorial Theory, Series B

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Section: Partial Factorization Of the Cover Polynomial Of Skyline Boardsmentioning

confidence: 82%

Factorization of the Cover Polynomial

Dworkin

1997

Journal of Combinatorial Theory, Series B

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“…The following theorem is stated (in a different, but equivalent, form) in [6], where it is claimed that it follows from the work of Brenti in [2]. Also, in [4] it is shown that the theorem follows from certain properties of the chromatic symmetric function of Stanley [9]. We give a proof that is different from both of these, but which better suits our purposes.…”

Section: Introductionmentioning

confidence: 86%

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Steingrı́msson¹

2001

Journal of Algebraic Combinatorics

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Abstract. Let G be a simple graph on d vertices. We define a monomial ideal K in the Stanley-Reisner ring A of the order complex of the Boolean algebra on d atoms. The monomials in K are in one-to-one correspondence with the proper colorings of G. In particular, the Hilbert polynomial of K equals the chromatic polynomial of G.The ideal K is generated by square-free monomials, so A/K is the Stanley-Reisner ring of a simplicial complex C. The h-vector of C is a certain transformation of the tail T (n) = n d − χ(n) of the chromatic polynomial χ of G. The combinatorial structure of the complex C is described explicitly and it is shown that the Euler characteristic of Cequals the number of acyclic orientations of G.

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“…There is also related work by R.P. Stanley [73] and T. Chow [18], and very recently, by P. Pitteloud [69].…”

Section: Graph Polynomialsmentioning

confidence: 94%

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

Makowsky

2007

Theory Comput Syst

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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities.

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The Path-Cycle Symmetric Function of a Digraph

Cited by 25 publications

References 20 publications

Factorization of the Cover Polynomial

Factorization of the Cover Polynomial

Untitled

From a Zoo to a Zoology: Towards a General Theory of Graph Polynomials

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