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The number of spanning trees in a new lexicographic product of graphs

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Abstract

The problem of counting the number of spanning trees is an old topic in graph theory with important applications to reliable network design. Usually, it is desirable to put forward a formula of the number of spanning trees for various graphs, which is not only interesting in its own right but also in practice. Since some large graphs can be composed of some existing smaller graphs by using the product of graphs, the number of spanning trees of such large graph is also closely related to that of the corresponding smaller ones. In this article, we establish a formula for the number of spanning trees in the lexicographic product of two graphs, in which one graph is an arbitrary graph G and the other is a complete multipartite graph. The results extend some of the previous work, which is closely related to the number of vertices and Lapalacian eigenvalues of smaller graphs only.

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References

  1. Cayley A. A theorem on trees. Quart J Math, 1889, 23: 276–278

    Google Scholar 

  2. Ho J, Hwang W. Wavelet Bayesian network image denoising. IEEE Trans Image Process, 2013, 22: 1277–1290

    Article  MathSciNet  Google Scholar 

  3. Ou W H, You X G, Cheung Y M, et al. Structured sparse coding for image representation based on L1-graph. ICPR, 2012: 3220–3223

    Google Scholar 

  4. Gonzalez J, Low Y, Guestrin C. Parallel splash belief propagation. J Mach Learn Res, 2009, 1: 1–48

    Google Scholar 

  5. Cheng C. Maximizing the total number spanning trees in a graph: two related problems in graph theory and optimization design theory. J Combin Theory B, 1981, 31: 240–248

    Article  MATH  Google Scholar 

  6. Li B L, Zhang S G. Heavy cycles and spanning trees with few leaves in weighted graphs. Appl Math Lett, 2011, 24: 908–910

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang Z, Li X. On the number of spanning trees of some composite graphs (in Chinese). Acta Math Sci, 1995, 15: 259–268

    MATH  Google Scholar 

  8. Li X, Huang Z. On the Number of Spanning Trees of Graphs and Applications of Networks. Harbin: Harbin Institute of Technology Press, 1999

    Google Scholar 

  9. Zhang Y, Yong X, Golin M J. The number of spanning trees in circulant graphs. Discrete Appl Math, 2000, 223: 337–350

    Article  MathSciNet  MATH  Google Scholar 

  10. Biggs N. Algebraic Graph Theory. 2nd ed. Cambridge: Cambridge University Press, 1993

    Google Scholar 

  11. Boesch F, Prodinger H. Spanning tree formulas and Chebyshev polynomials. Graph Combinator, 1986, 2: 191–200

    Article  MathSciNet  MATH  Google Scholar 

  12. Boesch F, Wang J F. A conjecture on the number of spanning trees in the square of a cycle. In: Notes from New York Graph Theory Day V. New York: New York Academy Sciences, 1982. 1–16

    Google Scholar 

  13. Chung K L, Yan W M. On the number of spanning trees of a multi-complete/star related graphs. Inform Process Lett, 2000, 76: 113–119

    Article  MathSciNet  Google Scholar 

  14. Gilbert B, Kelman A K. Laplacian spectra and spanning trees of threshold graphs. Discrete Math, 1996, 65: 255–273

    Article  Google Scholar 

  15. Imrich W, Izbicki H. Associative products of graphs. Monatsh Math, 1975, 80: 277–281

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang Z, Zhu Y F. Cyclic arc-connectivity in a Cartesian product digraph. Appl Math Lett, 2010, 23: 796–800

    Article  MathSciNet  MATH  Google Scholar 

  17. Li F, Peng Y, Zhao H X. On the number of spanning trees of the multi-lexicographic product of networks (in Chinese). Software, 2011, 7: 51–53

    Google Scholar 

  18. Li F, Xu Z B, Zhao H X, et al. On the number of spanning trees of the lexicographic product of networks. Scientia Sin Inform, 2012, 42: 949–959

    Article  Google Scholar 

  19. Li F, Wang W, Xu Z B, et al. Some results on the lexicographic product of vertex-transitive graphs. Appl Math Lett, 2011, 24: 1924–1926

    Article  MathSciNet  MATH  Google Scholar 

  20. Yegnanarayanan V, Thiripurasundari P R. On some graph operations and related applications. Electron Notes Discrete Math, 2009, 33: 123–130

    Article  MathSciNet  Google Scholar 

  21. Moon J. Enumerating Labelled Trees, Graph Theory and Theoretical Physics. New York: New York Academic Press, 1967. 261–272

    Google Scholar 

  22. Kelman A K, Chelnokov V M. A certain polynomials of a graph and graphs with an extremal number of trees. J Combin Theory B, 1974, 16: 197–214

    Article  Google Scholar 

  23. Wang W, Li F, Lu H L, et al. Graphs determined by their generalized characteristic polynomials. Linear Algebra Appl, 2011, 434: 1378–1387

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute of Information and System Sciences, School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, China

    Dong Liang, Feng Li & ZongBen Xu

  2. College of Computer Science, Qinghai Normal University, Xining, 810003, China

    Feng Li

  3. Ministry of Education key Lab for Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, 710049, China

    Dong Liang, Feng Li & ZongBen Xu

Authors
  1. Dong Liang
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  2. Feng Li
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  3. ZongBen Xu
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Correspondence to ZongBen Xu.

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Liang, D., Li, F. & Xu, Z. The number of spanning trees in a new lexicographic product of graphs. Sci. China Inf. Sci. 57, 1–9 (2014). https://doi.org/10.1007/s11432-014-5110-z

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  • DOI: https://doi.org/10.1007/s11432-014-5110-z

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