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A generalization of the Whitney rank generating function

Published online by Cambridge University Press:  24 October 2008

G. E. Farr
G. E. Farr
Affiliation:
CMR Group, Communications Division, ERL, Defence Science and Technology Organization, P.O. Box 4924, Kingston, A.C.T. 2604, Australia
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Abstract

The Whitney quasi-rank generating function, which generalizes the Whitney rank generating function (or Tutte polynomial) of a graph, is introduced. It is found to include as special cases the weight enumerator of a (not necessarily linear) code, the percolation probability of an arbitrary clutter and a natural generalization of the chromatic polynomial. The crucial construction, essentially equivalent to one of Kung, is a means of associating, to any function, a rank-like function with suitable properties. Some of these properties, including connections with the Hadamard transform, are discussed.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 2 , March 1993 , pp. 267 - 280
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Biggs, N. L.. Algebraic Graph Theory (Cambridge University Press, 1974).CrossRefGoogle Scholar
[2]Brylawski, T. H. and Oxley, J. G.. The Tutte polynomial and its applications. In Matroid Theory, vol. 3 (Cambridge University Press, to appear).Google Scholar
[3]Crapo, H. H. and Rota, G.-C.. On the Foundations of Combinatorial Theory: Combinatorial Geometries (M.I.T. Press, 1970).Google Scholar
[4]Farr, G. E.. A correlation inequality involving stable set and chromatic polynomials. J. Combin. Theory Ser. B, to appear.Google Scholar
[5]Greene, C.. Weight enumeration and the geometry of linear codes. Stud. Appl. Math. 55 (1976), 119–128.CrossRefGoogle Scholar
[6]Jaeger, F., Vertigan, D. L. and Welsh, D. J. A.. On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108 (1990), 36–53.CrossRefGoogle Scholar
[7]Kung, J. P. S.. The Rédei function of a relation. J. Combin. Theory Ser. A 29 (1980), 287–296.CrossRefGoogle Scholar
[8]Kung, J. P. S., Murty, M. R. and Rota, G.-C.. On the Rédei zeta function. J. Number Theory 12 (1980), 421–436.CrossRefGoogle Scholar
[9]MacWilliams, F. J. and Sloane, N. J. A.. The Theory of Error-Correcting Codes (North-Holland, 1977).Google Scholar
[10]MacWilliams, F. J., Sloane, N. J. A. and Goethals, J.-M.. The MacWilliams identities for nonlinear codes. Bell Syst. Tech. J. 51 (1972), 803–819.CrossRefGoogle Scholar
[11]Oxley, J. G. and Welsh, D. J. A.. On some percolation results of J. M. Hammersley. J. Appl. Probab. 16 (1979), 526–540.CrossRefGoogle Scholar
[12]Oxley, J. G. and Welsh, D. J. A.. The Tutte polynomial and percolation. In Graph Theory and Related Topics (Academic Press, 1979), pp. 329–339.Google Scholar
[13]Oxley, J. G. and Whittle, G. P.. A characterization of Tutte invariants of 2-polymatroids. J. Combin. Theory Ser. B, to appear.Google Scholar
[14]Tutte, W. T.. A ring in graph theory. Proc. Cambridge Philos. Soc. 43 (1947), 26–40.CrossRefGoogle Scholar
[15]Tutte, W. T.. A contribution to the theory of chromatic polynomials. Canad. J. Math. 6 (1954), 80–91.CrossRefGoogle Scholar
[16]Tutte, W. T.. On dichromatic polynomials. J. Combin. Theory 2 (1967), 301–320.CrossRefGoogle Scholar
[17]Welsh, D. J. A.. Matroid Theory. London Math. Soc. Monograph no. 8 (Academic Press, 1976).Google Scholar
[18]Whittle, G. P.. The critical problem for 2-polymatroids. Quart. J. Math. Oxford Ser. (2), to appear.Google Scholar
[19]Whittle, G. P.. Characteristic polynomials of weighted lattices. Adv. Math., to appear.Google Scholar