Startseite Mathematik Recurrences for the genus polynomials of linear sequences of graphs
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Recurrences for the genus polynomials of linear sequences of graphs

  • Yichao Chen EMAIL logo , Jonathan L. Gross , Toufik Mansour und Thomas W. Tucker
Veröffentlicht/Copyright: 23. Mai 2020
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Mathematica Slovaca
Aus der Zeitschrift Mathematica Slovaca Band 70 Heft 3

Abstract

Given a finite graph H, the nth member Gn of an H-linear sequence is obtained recursively by attaching a disjoint copy of H to the last copy of H in Gn−1 by adding edges or identifying vertices, always in the same way. The genus polynomial ΓG(z) of a graph G is the generating function enumerating all orientable embeddings of G by genus. Over the past 30 years, most calculations of genus polynomials ΓGn(z) for the graphs in a linear family have been obtained by partitioning the embeddings of Gn into types 1, 2, …, k with polynomials ΓGnj (z), for j = 1, 2, …, k; from these polynomials, we form a column vector Vn(z)=[ΓGn1(z),ΓGn2(z),…,ΓGnk(z)]t that satisfies a recursion Vn(z) = M(z)Vn−1(z), where M(z) is a k × k matrix of polynomials in z. In this paper, the Cayley-Hamilton theorem is used to derive a kth degree linear recursion for Γn(z), allowing us to avoid the partitioning, thereby yielding a reduction from k2 multiplications of polynomials to k such multiplications. Moreover, that linear recursion can facilitate proofs of real-rootedness and log-concavity of the polynomials. We illustrate with examples.

MSC 2010: Primary: 05C10
Keywords: genus polynomial; log-concavity

  1. Communicated by Peter Horák

References

[Bre89] Brenti, F.: Unimodal, log-concave, and Pőlya frequency sequences, Mem. Amer. Math. Soc. 413 (1989).10.1090/memo/0413Suche in Google Scholar

[CG93] Chen, J.—Gross, J. L.: Kuratowski-type theorems for average genus, J. Combin. Theory Ser. B 57 (1993), 100–121.10.1006/jctb.1993.1009Suche in Google Scholar

[CL10] Chen, Y.—Liu, Y.: On a conjecture of S. Stahl, Canad. J. Math. 62 (2010), 1058–1059.10.4153/CJM-2010-058-4Suche in Google Scholar

[FGS89] Furst, M.—Gross, J. L.—Statman, R.: Genus distributions for two classes of graphs, J. Combin. Theory Ser. B 46 (1989), 22–36.10.1016/0095-8956(89)90004-XSuche in Google Scholar

[Gold58] Goldberg, S.: Introduction to the Difference Equation, John Wiley and Sons, Inc., 1958.Suche in Google Scholar

[Gro14] Gross, J. L.: Embeddings of graphs of fixed treewidth and bounded degree, Ars Math. Contemp. 7 (2014), 379–403.10.26493/1855-3974.366.dd1Suche in Google Scholar

[GF87] Gross, J. L.—Furst, M. L.: Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11 (1987), 205–220.10.1002/jgt.3190110211Suche in Google Scholar

[GKMT18] Gross, J. L.—Khan, I. F.—Mansour, T.—Tucker, T. W.: Calculating genus polynomials via string operations and matrices, Ars Math. Contemp. 15 (2018), 267–295; Presented at the Eighth Slovenian Conference on Graph Theory, Kranjska Gora, Slovenia, 2015.10.26493/1855-3974.939.77dSuche in Google Scholar

[GKP10] Gross, J. L.—Khan, I. F.—Poshni, M. I.: Genus distribution of graph amalgamations: Pasting at root-vertices, Ars Combin. 94 (2010), 33–53.Suche in Google Scholar

[GKP14] Gross, J. L.—Khan, I. F.—Poshni, M. I.: Genus distributions for iterated claws, Electron. J. Combin. 21 (2014), #P1.1210.37236/2278Suche in Google Scholar

[GKR93] Gross, J. L.—Klein, E. W.—Rieper, R. G.: On the average genus of graphs, Graphs and Combinatorics 9 (1993), 153–162.10.1007/BF02988301Suche in Google Scholar

[GMT14] Gross, J. L.—Mansour, T.—Tucker, T. W.: Log-concavity of genus distributions of ring-like families of graphs, European J. Combin. 42 (2014), 74–91.10.1016/j.ejc.2014.05.008Suche in Google Scholar

[GMTW15] Gross, J. L.—Mansour, T.—Tucker, T. W.—Wang, D. G. L.: Log-concavity of combinations of sequences and applications to genus distributions, SIAM J. Discrete Math. 29 (2015), 1002–1029.10.1137/140978867Suche in Google Scholar

[GMTW16a] Gross, J. L.—Mansour, T.—Tucker, T. W.— Wang, D. G. L.: Root geometry of polynomial sequences I: Type (0, 1), J. Math. Anal. Appl. 433 (2016), 1261–1289.10.1016/j.jmaa.2015.08.039Suche in Google Scholar

[GMTW16b] Gross, J. L.—Mansour, T.—Tucker, T. W.— Wang, D. G. L.: Root geometry of polynomial sequences II: Type (1, 0), J. Math. Anal. Appl. 441 (2016), 499–528.10.1016/j.jmaa.2016.04.033Suche in Google Scholar

[GRT89] Gross, J. L.—Robbins, D. P.—Tucker, T. W.: Genus distributions for bouquets of circles, J. Combin. Theory (B) 47 (1989), 292–306.10.1016/0095-8956(89)90030-0Suche in Google Scholar

[GT87] Gross, J. L.—Tucker, T. W.: Topological Graph Theory, Dover, 2001; (orig. edition Wiley, 1987).Suche in Google Scholar

[Hog14] Hogben, L.: Handbook of Linear Algebra, second edition, CRC Press, 2014.10.1201/b16113Suche in Google Scholar

[HuKa12] Huh, J.—Katz, E.: Log-concavity of characteristic polynomials and the Bergman fan of matroids, Math. Ann. 354 (2012), 1103–1116.10.1007/s00208-011-0777-6Suche in Google Scholar

[Kun74] Kundu, S.: Bounds on the number of disjoint spanning trees, J. Combin. Ser. B 17 (1974), 199–203.10.1016/0095-8956(74)90087-2Suche in Google Scholar

[Las02] Lasserre, J. B.: Polynomials with all zeroes real and in a prescribed interval, J. Algebraic Combin. 16 (2002), 231–23710.1023/A:1021848304877Suche in Google Scholar

[LW07] Liu, L. L.—Wang, Y.: A unified approach to polynomial sequences with only real zeros, Adv. Appl. Math. 38 (2007), 542–560.10.1016/j.aam.2006.02.003Suche in Google Scholar

[PKG10] Poshni, M. I.—Khan, I. F.—Gross, J. L.: Genus distribution of graphs under edge-amalgamations, Ars Math. Contemp. 3 (2010), 69–86.10.26493/1855-3974.110.6b6Suche in Google Scholar

[PKG14] Poshni, M. I.—Khan, I. F.—Gross, J. L.: Genus distribution of iterated 3-wheels and 3-prisms, Ars Math. Contemp. 7 (2014), 423–440.10.26493/1855-3974.381.364Suche in Google Scholar

[Stah91] Stahl, S.: Permutation-partition pairs III: Embedding distributions of linear families of graphs, J. Combin. Theory (B) 52 (1991), 191–218.10.1016/0095-8956(91)90062-OSuche in Google Scholar

[Stah97] Stahl, S.: On the zeros of some genus polynomials, Canad. J. Math. 49 (1997), 617–640.10.4153/CJM-1997-029-5Suche in Google Scholar

[St86] Stanley, R. P.: Enumerative Combinatorics, Vol. 1, Wadsworth & Brooks/Cole, 1986.10.1007/978-1-4615-9763-6Suche in Google Scholar

[St89] Stanley, R. P.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Annals of the NY Acad. of Sci. 576 (1989), 500–535.10.1111/j.1749-6632.1989.tb16434.xSuche in Google Scholar

[Xu79] Xuong, N. H.: How to determine the maximum genus of a graph, J. Combin. Ser. B 26 (1979), 217–225.10.1016/0095-8956(79)90058-3Suche in Google Scholar

Received: 2019-04-22
Accepted: 2019-11-21
Published Online: 2020-05-23
Published in Print: 2020-06-25

© 2020 Mathematical Institute Slovak Academy of Sciences

Artikel in diesem Heft

  1. Regular papers
  2. Recurrences for the genus polynomials of linear sequences of graphs
  3. The existence of states on EQ-algebras
  4. On sidon sequences of farey sequences, square roots and reciprocals
  5. Repdigits as sums of three balancing numbers
  6. Lipschitz one sets modulo sets of measure zero
  7. Refinement of fejér inequality for convex and co-ordinated convex functions
  8. An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
  9. Coefficients problems for families of holomorphic functions related to hyperbola
  10. Triebel-Lizorkin capacity and hausdorff measure in metric spaces
  11. The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
  12. On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
  13. Density of summable subsequences of a sequence and its applications
  14. Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
  15. A note on cosine series with coefficients of generalized bounded variation
  16. Some geometric properties of the non-Newtonian sequence spaces lp(N)
  17. On sequence spaces defined by the domain of a regular tribonacci matrix
  18. Jointly separating maps between vector-valued function spaces
  19. Some fixed point theorems for multi-valued mappings in graphical metric spaces
  20. Monotone transformations on the cone of all positive semidefinite real matrices
  21. Locally defined operators in the space of Ck,ω-functions
  22. Some properties of D-weak operator topology
  23. Estimating the distribution of a stochastic sum of IID random variables
  24. An internal characterization of complete regularity
https://doi.org/10.1515/ms-2017-0368
Schlagwörter für diesen Artikel
genus polynomial; log-concavity

Artikel in diesem Heft

  1. Regular papers
  2. Recurrences for the genus polynomials of linear sequences of graphs
  3. The existence of states on EQ-algebras
  4. On sidon sequences of farey sequences, square roots and reciprocals
  5. Repdigits as sums of three balancing numbers
  6. Lipschitz one sets modulo sets of measure zero
  7. Refinement of fejér inequality for convex and co-ordinated convex functions
  8. An extension of q-starlike and q-convex error functions endowed with the trigonometric polynomials
  9. Coefficients problems for families of holomorphic functions related to hyperbola
  10. Triebel-Lizorkin capacity and hausdorff measure in metric spaces
  11. The method of upper and lower solutions for integral boundary value problem of semilinear fractional differential equations with non-instantaneous impulses
  12. On a system of three difference equations of higher order solved in terms of Lucas and Fibonacci numbers
  13. Density of summable subsequences of a sequence and its applications
  14. Rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space
  15. A note on cosine series with coefficients of generalized bounded variation
  16. Some geometric properties of the non-Newtonian sequence spaces lp(N)
  17. On sequence spaces defined by the domain of a regular tribonacci matrix
  18. Jointly separating maps between vector-valued function spaces
  19. Some fixed point theorems for multi-valued mappings in graphical metric spaces
  20. Monotone transformations on the cone of all positive semidefinite real matrices
  21. Locally defined operators in the space of Ck,ω-functions
  22. Some properties of D-weak operator topology
  23. Estimating the distribution of a stochastic sum of IID random variables
  24. An internal characterization of complete regularity
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 15.12.2025 von https://www.degruyterbrill.com/document/doi/10.1515/ms-2017-0368/pdf
Button zum nach oben scrollen