Skip to main content

Advertisement

SpringerLink
Log in

Inequalities for the Number of Walks in Graphs

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We investigate the growth of the number w k of walks of length k in undirected graphs as well as related inequalities. In the first part, we deduce the inequality w 2a+c ⋅w 2(a+b)+c ≤w 2a ⋅w 2(a+b+c), which we call the Sandwich Theorem. It unifies and generalizes an inequality by Lagarias et al. and an inequality by Dress and Gutman. In the same way, we derive the inequality w 2a+c (v,v)⋅w 2(a+b)+c (v,v)≤w 2a (v,v)⋅w 2(a+b+c)(v,v) for the number w k (v,v) of closed walks of length k starting at a given vertex v. We then use a theorem of Blakley and Dixon to show \(w_{2\ell+p}^{k}\leq w_{2\ell+pk}\cdot w_{2\ell}^{k-1}\), which unifies and generalizes an inequality by Erdős and Simonovits and, again, the inequality by Dress and Gutman. Both results can be translated directly into the corresponding forms using the higher order densities, which extends former results.

In the second part, we provide a new family of lower bounds for the largest eigenvalue λ 1 of the adjacency matrix based on closed walks. We apply the Sandwich Theorem to show monotonicity in this and a related family of lower bounds of Nikiforov. This leads to generalized upper bounds for the energy of graphs.

In the third part, we demonstrate that a further natural generalization of the Sandwich Theorem is not valid for general graphs. We show that the inequality w a+b ⋅w a+b+c ≤w a ⋅w a+2b+c does not hold even in very restricted cases like w 1⋅w 2≤w 0⋅w 3 (i.e., \(\bar{d}\cdot w_{2}\leq w_{3}\)) in the context of bipartite or cycle free graphs. In contrast, we show that surprisingly this inequality is always satisfied for trees and we show how to construct worst-case instances (regarding the difference of both sides of the inequality) for a given degree sequence. We also prove the inequality w 1⋅w 4≤w 0⋅w 5 (i.e., \(\bar{d}\cdot w_{4}\leq w_{5}\)) for trees and conclude with a corresponding conjecture for longer walks.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Notes

  1. Note that Nikiforov used odd values for k which is due to the fact that he counted vertices instead of edges for defining w k .

References

  1. Ahlswede, R., Katona, G.O.H.: Graphs with maximal number of adjacent pairs of edges. Acta Math. Acad. Sci. Hung. 32(1–2), 97–120 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Feige, U., Wigderson, A., Zuckerman, D.: Derandomized graph products. Comput. Complex. 5(1), 60–75 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blakley, G.R., Dixon, R.D.: Hölder type inequalities in cones. J. Math. Anal. Appl. 14(1), 1–4 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blakley, G.R., Roy, P.: A Hölder type inequality for symmetric matrices with nonnegative entries. Proc. Am. Math. Soc. 16(6), 1244–1245 (1965)

    MathSciNet  MATH  Google Scholar 

  5. Chakrabarti, D., Wang, Y., Wang, C., Leskovec, J., Faloutsos, C.: Epidemic thresholds in real networks. ACM Trans. Inf. Syst. Secur. 10(4), 13:1–13:26 (2008)

    Article  Google Scholar 

  6. Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997)

    MATH  Google Scholar 

  7. Cioabă, S.M.: Some applications of eigenvalues of graphs. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 357–379. Birkhäuser, Basel (2011), Chap. 14

    Chapter  Google Scholar 

  8. Collatz, L., Sinogowitz, U.: Spektren endlicher Grafen. Abh. Math. Semin. Univ. Hamb. 21(1), 63–77 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cvetković, D.M.: The generating function for variations with restrictions and paths of the graph and self-complementary graphs. Univ. Beog., Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 320–328(322), 27–34 (1970)

    Google Scholar 

  10. Cvetković, D.M.: Graphs and their spectra. Univ. Beog., Publ. Elektrotehn. Fak., Ser. Mat. Fiz. 354–456(354), 1–50 (1971)

    Google Scholar 

  11. Cvetković, D.M.: Applications of graph spectra: An introduction to the literature. In: Applications of Graph Spectra. Zbornik Radova, vol. 13(21), pp. 7–31. Mathematical Institute SANU, Belgrade (2009)

    Google Scholar 

  12. Cvetković, D.M., Doob, M., Gutman, I., Torgašev, A.: Recent Results in the Theory of Graph Spectra. Annals of Discrete Mathematics, vol. 36. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  13. Cvetković, D.M., Doob, M., Sachs, H.: Spectra of Graphs—Theory and Applications. Deutscher Verlag der Wissenschaften, Berlin (1979)

    Google Scholar 

  14. Cvetković, D.M., Rowlinson, P.: The largest eigenvalue of a graph: A survey. Linear Multilinear Algebra 28(1), 3–33 (1990)

    Article  MATH  Google Scholar 

  15. Cvetković, D.M., Rowlinson, P., Simić, S.K.: Eigenspaces of Graphs. Encyclopedia of Mathematics and Its Applications, vol. 66. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  16. De Caen, D.: An upper bound on the sum of squares of degrees in a graph. Discrete Math. 185, 245–248 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dress, A., Gutman, I.: The number of walks in a graph. Appl. Math. Lett. 16(5), 797–801 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  18. Erdős, P., Simonovits, M.: Compactness results in extremal graph theory. Combinatorica 2(3), 275–288 (1982)

    Article  MathSciNet  Google Scholar 

  19. Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fiol, M.À., Garriga, E.: Number of walks and degree powers in a graph. Discrete Math. 309(8), 2613–2614 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Gutman, I.: The energy of a graph: Old and new results. In: Proceedings of the Euroconference Algebraic Combinatorics and Applications (ALCOMA’99), pp. 196–211. Springer, Berlin (2001)

    Google Scholar 

  22. Hansen, P., Vukičević, D.: Comparing the Zagreb indices. Croat. Chem. Acta 80(2), 165–168 (2007)

    Google Scholar 

  23. Harary, F., Schwenk, A.J.: The spectral approach to determining the number of walks in a graph. Pac. J. Math. 80(2), 443–449 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hemmecke, R., Kosub, S., Mayr, E.W., Täubig, H., Weihmann, J.: Inequalities for the number of walks in trees and general graphs and a generalization of a theorem of Erdős and Simonovits. Technical Report TUM-I1109, Department of Computer Science, Technische Universität München (2011)

  25. Hemmecke, R., Kosub, S., Mayr, E.W., Täubig, H., Weihmann, J.: Inequalities for the number of walks in graphs. In: Proceedings of the 9th Meeting on Analytic Algorithmics and Combinatorics (ANALCO’12), pp. 26–39. SIAM, Philadelphia (2012)

    Google Scholar 

  26. Hoffman, A.J.: Three observations on nonnegative matrices. J. Res. Natl. Bur. Stand. B, Math. Math. Phys. 71(1), 39–41 (1967)

    Article  MATH  Google Scholar 

  27. Hoffman, A.J.: On eigenvalues and colorings of graphs. In: Harris, B. (ed.) Graph Theory and Its Applications, pp. 79–91. Academic Press, San Diego (1970)

    Google Scholar 

  28. Hofmeister, M.: Spectral radius and degree sequence. Math. Nachr. 139(1), 37–44 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hofmeister, M.: A note on almost regular graphs. Math. Nachr. 166(1), 259–262 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hong, Y., Zhang, X.D.: Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees. Discrete Math. 296(2–3), 187–197 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hou, Y., Tang, Z., Woo, C.: On the spectral radius, k-degree and the upper bound of energy in a graph. MATCH Commun. Math. Comput. Chem. 57(2), 341–350 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Hu, S.: A sharp lower bound of the spectral radius of simple graphs. Appl. Anal. Discrete Math. 3(2), 379–385 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ilić, A., Stevanović, D.: On comparing Zagreb indices. MATCH Commun. Math. Comput. Chem. 62(3), 681–687 (2009)

    MathSciNet  MATH  Google Scholar 

  34. Kosub, S.: Local density. In: Brandes, U., Erlebach, T. (eds.) Network Analysis—Methodological Foundations. LNCS, vol. 3418, pp. 112–142. Springer, Berlin (2005)

    Chapter  Google Scholar 

  35. Lagarias, J.C., Mazo, J.E., Shepp, L.A., McKay, B.D.: An inequality for walks in a graph. SIAM Rev. 25(3), 403 (1983)

    Article  Google Scholar 

  36. Lagarias, J.C., Mazo, J.E., Shepp, L.A., McKay, B.D.: An inequality for walks in a graph. SIAM Rev. 26(4), 580–582 (1984)

    Article  Google Scholar 

  37. Lewis, H.R., Papadimitriou, C.H.: Symmetric space-bounded computation. Theor. Comput. Sci. 19(2), 161–187 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  38. London, D.: Inequalities in quadratic forms. Duke Math. J. 33(3), 511–522 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  39. London, D.: Two inequalities in nonnegative symmetric matrices. Pac. J. Math. 16(3), 515–536 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  40. Marcus, M., Newman, M.: The sum of the elements of the powers of a matrix. Pac. J. Math. 12(2), 627–635 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  41. McClelland, B.J.: Properties of the latent roots of a matrix: The estimation of π-electron energies. J. Chem. Phys. 54(2), 640–643 (1971)

    Article  Google Scholar 

  42. Mulholland, H.P., Smith, C.A.B.: An inequality arising in genetical theory. Am. Math. Mon. 66(8), 673–683 (1959)

    Article  MathSciNet  Google Scholar 

  43. Mulholland, H.P., Smith, C.A.B.: Corrections: An inequality arising in genetical theory. Am. Math. Mon. 67(2), 161 (1960)

    Article  Google Scholar 

  44. Nikiforov, V.: Walks and the spectral radius of graphs. Linear Algebra Appl. 418(1), 257–268 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Nikiforov, V.: The sum of the squares of degrees: Sharp asymptotics. Discrete Math. 307(24), 3187–3193 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. Nosal, E.: Eigenvalues of graphs. Master’s thesis, University of Calgary (1970)

  47. Peled, U.N., Petreschi, R., Sterbini, A.: (n,e)-graphs with maximum sum of squares of degrees. J. Graph Theory 31(4), 283–295 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  48. Van Mieghem, P.: Graph Spectra for Complex Networks. Cambridge University Press, Cambridge (2011)

    MATH  Google Scholar 

  49. Vukičević, D., Graovac, A.: Comparing Zagreb M 1 and M 2 indices for acyclic molecules. MATCH Commun. Math. Comput. Chem. 57(3), 587–590 (2007)

    MathSciNet  MATH  Google Scholar 

  50. Wang, H.: Extremal trees with given degree sequence for the Randić index. Discrete Math. 308(15), 3407–3411 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  51. Wilf, H.S.: The eigenvalues of a graph and its chromatic number. J. Lond. Math. Soc. 42, 330–332 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  52. Wilf, H.S.: Spectral bounds for the clique and independence numbers of graphs. J. Comb. Theory, Ser. B 40(1), 113–117 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  53. Yu, A., Lu, M., Tian, F.: On the spectral radius of graphs. Linear Algebra Appl. 387, 41–49 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We want to thank Daniel Fleischer, Alexander Offtermatt-Souza, Moritz Maaß, Riko Jacob, and Holger Täubig for valuable remarks and discussions.

Author information

Authors and Affiliations

  1. Institut für Informatik, TU München, Boltzmannstr. 3, 85748, Garching, Germany

    Hanjo Täubig, Jeremias Weihmann & Ernst W. Mayr

  2. Fachbereich Informatik und Informationswissenschaft, Universität Konstanz, Fach 67, 78457, Konstanz, Germany

    Sven Kosub

  3. Institut für Mathematik, TU München, Boltzmannstr. 3, 85748, Garching, Germany

    Raymond Hemmecke

Authors
  1. Hanjo Täubig
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeremias Weihmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sven Kosub
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Raymond Hemmecke
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ernst W. Mayr
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanjo Täubig.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Täubig, H., Weihmann, J., Kosub, S. et al. Inequalities for the Number of Walks in Graphs. Algorithmica 66, 804–828 (2013). https://doi.org/10.1007/s00453-013-9766-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-013-9766-3

Keywords

Search

Navigation