Note on the Subgraph Component Polynomial

We establish a broad generalization of Whitney's broken circuit theorem on the chromatic polynomial of a graph to sums of the type A⊆S f (A) where S is a finite set and f is a mapping from the power set of S to an abelian group. We give applications to the domination polynomial and the subgraph component polynomial of a graph, the chromatic polynomial of a hypergraph, the characteristic polynomial and Crapo's beta invariant of a matroid, and the principle of inclusion-exclusion. Thus, we discover several known and new results in a concise and unified way. As further applications of our main result, we derive a new generalization of the maximums-minimums identity and of a theorem due to Blass and Sagan on the Möbius function of a finite lattice, which generalizes Rota's crosscut theorem. For the classical Möbius function, both Euler's totient function and its Dirichlet inverse, and the reciprocal of the Riemann zeta function we obtain new expansions involving the greatest common divisor resp. least common multiple. We finally establish an even broader generalization of Whitney's broken circuit theorem in the context of convex geometries (antimatroids).

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“…Eq. (19) can alternatively be deduced from Eq. (18), which is due to Heron [18], by considering the derivative of χ(M, x) at x = 1.…”

Section: Characteristic Polynomial and Beta Invariantmentioning

confidence: 99%

“…This polynomial has seen applications in social network analysis [3] and formal language theory [6]. For some recent results on Q(G, x, y), the reader is referred to [19].…”

Section: Subgraph Component Polynomialmentioning

confidence: 99%

An Abstraction of Whitney's Broken Circuit Theorem

Dohmen

Trinks

2014

Electron. J. Combin.

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“…In [9], Narushima presented a cancellation for the inclusion-exclusion principle, depending on a prescribed ordering on the index set P. This result was later improved by Dohmen [2]. Using the same technique, Dohmen [5] also established an abstraction of Whitney's broken circuit theorem, which not only applies to the chromatic polynomial, but also to other graph polynomials, see [3,4,5,8,12] for details.…”

Section: Introductionmentioning

confidence: 99%

Inclusion–exclusion by ordering-free cancellation

Chen

Qian

2019

Journal of Combinatorial Theory, Series A

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Whitney's broken circuit theorem gives a graphical example to reduce the number of the terms in the sum of the inclusion-exclusion formula by a predicted cancellation. So far, the known cancellations for the formula strongly depend on the prescribed (linear or partial) ordering on the index set. We give a new cancellation method, which does not require any ordering on the index set. Our method extends all the 'ordering-based' methods known in the literatures and in general reduces more terms. As examples, we use our method to improve some relevant results on graph polynomials.

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Inclusion-exclusion by ordering-free cancellation

Chen¹,

Qian²

2018

Preprint

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Note on the Subgraph Component Polynomial

Cited by 3 publications

References 25 publications

An Abstraction of Whitney's Broken Circuit Theorem

An Abstraction of Whitney's Broken Circuit Theorem

Inclusion–exclusion by ordering-free cancellation

Inclusion-exclusion by ordering-free cancellation

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