Extension of Chromatic Polynomials by Utilizing Mobius Inversion Theorem

Classification of Algebraic Properties of Chromatic Polynomials

Journal of Scientific Research ◽

10.3329/jsr.v5i3.11634 ◽

2013 ◽

Vol 5 (3) ◽

pp. 469-477 ◽

Cited By ~ 2

Author(s):

R. V. N. S. Rao ◽

J. V. Rao

Keyword(s):

Chromatic Polynomial ◽

Möbius Function ◽

Chromatic Polynomials ◽

Inversion Theorem ◽

Mobius Function ◽

Möbius Inversion ◽

Algebraic Properties

This manuscript attempts to introduce the concept of chromatic polynomials of total graphs using Mobius inversion theorem. In fact it studies various algebraic properties of chromatic polynomial using Mobius inversion theorem. Keywords: Bond lattice; Chromatic polynomial; Mobius function; Poset. © 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved. doi: http://dx.doi.org/10.3329/jsr.v5i3.11634 J. Sci. Res. 5 (3), 469-477 (2013)

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Erratum: Carlsson-Gelatt-Ehrenreich technique and the Möbius inversion theorem

Physical Review B ◽

10.1103/physrevb.47.593.3 ◽

1993 ◽

Vol 47 (1) ◽

pp. 593-593 ◽

Cited By ~ 1

Author(s):

Nan-xian Chen ◽

Guang-bao Ren

Keyword(s):

Inversion Theorem ◽

Möbius Inversion

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On an inversion theorem of Möbius

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002187x ◽

1980 ◽

Vol 30 (1) ◽

pp. 15-32 ◽

Cited By ~ 8

Author(s):

J. H. Loxton ◽

J. W. Sanders

Keyword(s):

Nineteenth Century ◽

Numerical Integration ◽

Inversion Theorem ◽

Möbius Inversion ◽

Inversion Principle ◽

Infinite Sums

AbstractThe theme of the paper is a Möbius inversion principle for infinite sums. We deal with the origins and unprincipled use of this idea in the nineteenth century, its rigorous justification under minimal hypotheses and some applications to a problem in numerical integration.

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Chen-Möbius inversion theorem and a structural representation of crystallographic direction families

Physics Letters A ◽

10.1016/0375-9601(93)90358-7 ◽

1993 ◽

Vol 184 (1) ◽

pp. 119-126 ◽

Cited By ~ 7

Author(s):

Qian Xie ◽

Mei-chun Huang

Keyword(s):

Crystallographic Direction ◽

Structural Representation ◽

Inversion Theorem ◽

Möbius Inversion

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Some Characterizations of the Systems Represented by Choquet and Multi-Linear Functionals through the use of Möbius Inversion

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488597000427 ◽

1997 ◽

Vol 05 (05) ◽

pp. 547-561 ◽

Cited By ~ 14

Author(s):

Katsushige Fujimoto ◽

Toshiaki Murofushi

Keyword(s):

Structural Analysis ◽

Fuzzy Measure ◽

Fuzzy Integral ◽

Linear Functionals ◽

Inversion Theorem ◽

Möbius Inversion ◽

Integral Systems

The systems represented by the Choquet or the multi-linear fuzzy integral with respect to fuzzy measure is equivalently decomposable into hierarchically sub-systems through the use of Inclusion-Exclusion Covering (IEC) (Theorem 4.1,4.2). Hence, IEC is one of very useful concepts/indexes for structural analysis of the fuzzy integral systems (short for: the systems represented by the Choquet or the multi-linear fuzzy integral) However, it is quite difficult to identify all IEC's. This paper shows a method for identifying it easily, through the use of Möbius inversion (Theorem 5.1).

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On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

The Electronic Journal of Combinatorics ◽

10.37236/647 ◽

2011 ◽

Vol 18 (1) ◽

Cited By ~ 1

Author(s):

Jacob A. White

Keyword(s):

Chromatic Polynomial ◽

Partition Lattice ◽

Chromatic Polynomials ◽

Möbius Inversion

In this paper we introduce multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, Pönitz and Tittman. We prove that specializations of these new polynomials recover polynomials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs, all weighted according to certain statistics. We also prove that the polynomials can be defined in terms of Möbius inversion on the partition lattice of a hypergraph, and we compute these polynomials for various classes of hypergraphs. We also consider trivariate polynomials, which we call the hyperedge elimination polynomial and the trivariate chromatic polynomial.

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