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On the (Signless) Laplacian Permanental Polynomials of Graphs

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Abstract

Let G be a graph, and let L(G) and Q(G) denote respectively the Laplacian matrix and the signless Laplacian matrix of G. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L(G) (respectively, Q(G)). In this paper, we give combinatorial expressions for the first five coefficients of the (signless) Laplacian permanental polynomial. The characterizing properties of the (signless) Laplacian permanental polynomial are investigated and some graphs determined by the (signless) Laplacian permanental polynomial are presented. Furthermore, we compute the (signless) Laplacian permanental polynomials for all graphs on at most 10 vertices, and count the number of such graphs for which there is another graph with the same (signless) Laplacian permanental polynomial.

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References

  1. Botti, P., Merris, R., Vega, C.: Laplacian permanents of trees. SIAM J. Discrete Math. 5, 460–466 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brualdi, R.A., Goldwasser, J.L.: Permanent of the Laplacian matrix of trees and bipartite graphs. Discrete Math. 48, 1–21 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cash, G.G.: Permanental polynomials of smaller fullerenes. J. Chem. Inf. Comput. Sci. 40, 1207–1209 (2000)

    Article  Google Scholar 

  4. Cash, G.G., Gutman, I.: The Laplacian permanental polynomial: formulas and algorithms. MATCH Commun. Math. Comput. Chem. 51, 129–136 (2004)

    MathSciNet  MATH  Google Scholar 

  5. Faria, I.: Permanental roots and the star degree of a graph. Linear Algebra Appl. 64, 255–265 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Faria, I.: Multiplicity of integer roots of polynomials of graphs. Linear Algebra Appl. 229, 15–35 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Geng, X., Hu, X., Li, S.: Further results on permanental bounds for the Laplacian matrix of trees. Linear Multilinear Algebra 58, 571–587 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Geng, X., Hu, S., Li, S.: Permanental bounds of the Laplacian matrix of trees with given domination number. Graph Combin. 31, 1423–1436 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Godsil, C.D.: Algebraic Combinatorics. Chapman and Hall, New York (1993)

    MATH  Google Scholar 

  10. Goldwasser, J.L.: Permanent of the Laplacian matrix of trees with a given matching. Discrete Math. 61, 197–212 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gutman, I.: Relation between the Laplacian and the ordinary characteristic polynomial. MATCH Commun. Math. Comput. Chem. 47, 133–140 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Haemers, W.H., Spence, E.: Enumeration of cospectral graphs. Eur. J. Combin. 25, 199–211 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ishaq, M., Merris, R., Zaslawsky, E.: Problems concerning permanental polynomials of graphs. Linear Multilinear Algebra 15, 345–350 (1984)

    Article  Google Scholar 

  14. Kasum, D., Trinajstić, N., Gutman, I.: Chemical graph theory. III. On permanental polynomial. Croat. Chem. Acta 54, 321–328 (1981)

    Google Scholar 

  15. Li, S., Li, Y., Zhang, X.: Edge-grafting theorems on permanents of the Laplacian matrices of graphs and their applications. Electron. J. Linear Algebra 26, 28–48 (2013)

    MathSciNet  MATH  Google Scholar 

  16. Li, W., Liu, S., Wu, T., Zhang, H.: On the permanental polynomials of graphs. In: Shi, Y., Dehmer, M., Li, X., Gutman, I. (eds.) Graph Polynomials, pp. 101–122. CRC Press, Boca Raton (2017)

    Chapter  Google Scholar 

  17. Li, S., Zhang, L.: Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs. Linear Multilinear Algebra 59, 145–158 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, S., Zhang, L.: Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter \(d\). Graphs Combin. 28, 531–546 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, S., Zhang, H.: On the characterizing properties of the permanental polynomials of graphs. Linear Algebra Appl. 438, 157–172 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, X., Wu, T.: Computing the permanental polynomials of graphs. Appl. Math. Comput. 304, 103–113 (2017)

    MathSciNet  MATH  Google Scholar 

  21. McKay, B.D., Piperno, A.: Practical graph isomorphism. II. J. Symbolic Comput. 60, 94–112 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Merris, R.: The Laplacian permanental polynomial for trees. Czechslovak Math. J. 32, 397–403 (1982)

    MathSciNet  MATH  Google Scholar 

  23. Merris, R., Rebman, K.R., Watkins, W.: Permanental polynomials of graphs. Linear Algebra Appl. 38, 273–288 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  24. de Mier, A., Noy, M.: Tutte uniqueness of line graphs. Discrete Math. 301, 57–65 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tong, H., Liang, H., Bai, F.: Permanental polynomials of the larger fullerenes. MATCH Commun. Math. Comput. Chem. 56, 141–152 (2006)

    MathSciNet  MATH  Google Scholar 

  26. Valiant, L.G.: The complexity of computing the permanent. Theoret. Comput. Sci. 8, 189–201 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. Vrba, A.: Principal subpermanents of the Laplacian matrix. Linear Multilinear Algebra 19, 335–346 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11501050) and the Fundamental Research Funds for the Central Universities CHD (Grant Nos. 300102129109, 300102128201, 300102128104).

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  1. School of Science, Chang’an University, Xi’an, Shaanxi, 710064, People’s Republic of China

    Shunyi Liu

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  1. Shunyi Liu
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Liu, S. On the (Signless) Laplacian Permanental Polynomials of Graphs. Graphs and Combinatorics 35, 787–803 (2019). https://doi.org/10.1007/s00373-019-02033-2

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  • DOI: https://doi.org/10.1007/s00373-019-02033-2

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