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On the Zeros of Some Genus Polynomials

Published online by Cambridge University Press:  20 November 2018

Saul Stahl
Saul Stahl*
Affiliation:
Department of Mathematics, University of Kansas, Lawrence, KS, U.S.A. 66045 e-mail: stahl@kuhub.cc.ukans.edu
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Abstract

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In the genus polynomial of the graph G, the coefficient of xk is the number of distinct embeddings of the graph G on the oriented surface of genus k. It is shown that for several infinite families of graphs all the zeros of the genus polynomial are real and negative. This implies that their coefficients, which constitute the genus distribution of the graph, are log concave and therefore also unimodal. The geometric distribution of the zeros of some of these polynomials is also investigated and some new genus polynomials are presented.

Keywords

05C1005A1530C1526C10
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 49 , Issue 3 , 01 June 1997 , pp. 617 - 640
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Archdeacon, D., Calculations on the average genus and genus distribution of graphs, Congr. Numer. 67(1988), 114–124.Google Scholar
2. Archdeacon, D., The orientable genus is nonadditive, J. Graph Theory 10(1986), 385–402.Google Scholar
3. Archdeacon, D., The nonorientable genus is additive, J. Graph Theory 10(1986), 363–384.Google Scholar
4. Brenti, F., Royle, G.F. and Wagner, D.G., Location of zeros of chromatic and related polynomials of graphs, Canad. J. Math. 46(1994), 55-80.Google Scholar
5. Comtet, L., Advanced combinatorics, Reidel Publ. Co. Boston, 1974.Google Scholar
6. Cori, R., Machi, A., Penaud, J.G. and Vauquelin, B., On the automorphism group of a planar hypermap, European J. Combin. 2(1981), 331–334.Google Scholar
7. Duke, R.A., The genus, regional number, and Betti number of a graph, Canad. J.Math. 134(1966), 817–822.Google Scholar
8. Furst, M.L., Gross, J.L. and Statman, R., Genus distributions of two classes of graphs, J. Combinatorial Theory (B) 46(1989), 22–36.Google Scholar
9. Gross, J.L., Robbins, D.P. and Tucker, T., Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47(1989), 292–306.Google Scholar
10. Gross, J.L. and Tucker, T., Topological graph theory, Wiley-Interscience, New York, 1987.Google Scholar
11. Gross, J.L. and Furst, M., Hierarchy for imbedding distribution invariants of a graph, J. Graph Theory, 11(1987), 205–220.Google Scholar
12. Harer, J. and Zagier, D., The Euler characteristics of the moduli space of curves, Invent. Math. 85(1986), 457–485.Google Scholar
13. Heilman, O.J. and Lieb, E.H., Theory of monomer-dimer systems, Commun. Math. Phys. 25(1972), 190– 232.Google Scholar
14. Jackson, D.M., Counting cycles in permutations by group characters, with applications to a topological problem, Trans.Amer.Math. Soc. 299(1987), 785–801.Google Scholar
15. Keilson, J. and Gerber, H., Some results for discrete unimodality, J. Amer. Statist. Assoc. 66(1971), 386–389.Google Scholar
16. McGeoch, L.A., Genus distributions for circular and Möbius ladders, manuscript.Google Scholar
17. Rieper, R.G., Doctoral dissertation, Western Mischigan University, August 1987.Google Scholar
18. Stahl, S., Permutation-partition pairs: A combinatorial generalization of graph embeddings, Trans. Amer. Math. Soc. 259(1980), 129–145.Google Scholar
19. Stahl, S., Permuttion-partition pairs III: Embedding distributions of linear families of graphs, J. Combin. Theory Ser. B 52(1991), 191–218.Google Scholar
20. Stahl, S., On the number of maximum genus embeddings of almost all graphs, European J. Combin. 13(1992), 119–126.Google Scholar
21. Stahl, S., Region distributions of some small diameter graphs, Discrete Math. 89(1991), 281–299.Google Scholar
22. Stahl, S., A combinatorial analog of the Jordan Curve Theorem, J. Combin. Theory Ser. B 35(1983), 28–38.Google Scholar
23. Stahl, S., A duality for permutations, Discrete Math. 71(1988), 257–271.Google Scholar
24. Stanley, R.P., Log-concave and unimodal sequences in algebra, combinatorics, and geometry, In: Graph Theory and its Applications: East and West, Ann. New York Acad. Sci. 576(1986), 500–535.Google Scholar
25. Wagner, D.G., The partition polynomial of a finite set system, J. Combin. Theory Ser.A 56(1991), 138–159.Google Scholar
26. Wagner, D.G., Total positivity of Hadamard products, J. Math. Anal. Appl. 163(1992), 459–483.Google Scholar
27. Walkup, D.W., How many ways can a permutation be factored into two n-cycles, ? DiscreteMath. 28(1979), 315–319.Google Scholar