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On the normalized Laplacian spectral radii of a graph and its line graph

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Abstract

Normalized Laplacian eigenvalues are very popular in spectral graph theory. The normalized Laplacian spectral radius \(\rho _1(G)\) of a graph G is the largest eigenvalue of normalized Laplacian matrix of G. In this paper, we determine the extremal graph for the minimum normalized Laplacian spectral radii of nearly complete graphs. We present several lower bounds on \(\rho _1(G)\) in terms of graph parameters and characterize the extremal graphs. Still, there is no result on the normalized Laplacian eigenvalues of line graphs. Here, we obtain sharp lower bounds on the normalized Laplacian spectral radii of line graphs. Moreover, we compare \(\rho _1(G)\) and \(\rho _1(L_G)\) \((L_G \text{ is } \text{ the } \text{ line } \text{ graph } \text{ of } \)G) in some class of graphs as they are incomparable in the general case. Finally, we present a relation on the normalized Laplacian spectral radii of a graph and its line graph.

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References

  • Bondy JA, Murty USR (1976) Graph theory with applications. Macmillan Press, New York

    Book  Google Scholar 

  • Butler S (2008) Eigenvalues and structures of graphs. Ph.D. dissertation, University of California, San Diego

  • Cavers M (2010) The normalized Laplacian matrix and general Randić index of graphs. Ph.D. dissertation, University of Regina

  • Cai L, Ellis JA (1992) Edge colouring line graphs of unicyclic graphs. Discrete Appl Math 36:75–82

    Article  MathSciNet  Google Scholar 

  • Chung FK (1997) Spectral graph theory. American Mathematical Society, Providence

    MATH  Google Scholar 

  • Das KC, Sun S (2016) Extremal graph on normalized Laplacian spectral radius and energy. Electron J Linear Algebra 29:237–253

    Article  Google Scholar 

  • Foster RM (1949) The average impedance of an electrical network, In Contributions to Applied Mechanics (Reissner Aniversary Volume). Edwards Brothers, Ann Arbor, pp 333–340

    Google Scholar 

  • Guo JM, Li J, Shiu WC (2016) Effects on the normalized Laplacian spectral radius of non-bipartite graphs under perturbation and their applications. Linear Multilinear Algebra 64(11):2177–2187

    Article  MathSciNet  Google Scholar 

  • Guo JM, Li J, Shiu WC (2016) The largest normalized Laplacian spectral radius of non-bipartite graphs. Bull Malays Math Sci Soc 39(1):77–87

    Article  MathSciNet  Google Scholar 

  • Gutman I, Robbiano M, Martins EA, Cardoso DM, Medina L, Rojo O (2010) Energy of line graphs. Linear Algebra Appl 433:1312–1323

    Article  MathSciNet  Google Scholar 

  • Kelmans AK, Chelnokov VM (1974) A certain polynomial of a graph and graphs with an extremal number of trees. J Combin Theory B 16:197–214

    Article  MathSciNet  Google Scholar 

  • Klein DJ, Randić M (1993) Resistance distance. J Math Chem 12:81–95

    Article  MathSciNet  Google Scholar 

  • Li J, Guo J-M, Shiu WC (2014) Bounds on normalized Laplacian eigenvalues of graphs. J Inequalities Appl 316:1–8

    MathSciNet  MATH  Google Scholar 

  • Milovanović EI, Matejić MM, Milovanović IŽ (2019) On the normalized Laplacian spectral radius, Laplacian incidence energy and Kemeny’s constant. Linear Algebra Appl 582:181–196

    Article  MathSciNet  Google Scholar 

  • Milovanović EI, Milovanović IŽ (2015) Remarks on bounds of normalized Laplacian Eigenvalues of graphs. arXiv:1506.05762v1

  • Palacios JL (2004) Foster’s formulas via probability and the Kirchhoff Index. Methodol Comput Appl Probab 6:381–387

    Article  MathSciNet  Google Scholar 

  • Shi L (2007) Spectral radii of a graph and its line graph. Linear Algebra Appl 422:58–66

    Article  MathSciNet  Google Scholar 

  • Schur I (1923) über eine Klasse von Mittelbildungen mit Anwendungen auf die Determinantentheorie. Sitzungsber Berl Math Ges 22:9–20

    MATH  Google Scholar 

  • Stein WA (2015) Sage mathematics software (version 6.8). The Sage Development Team. http://www.sagemath.org

  • Stevanović D, Gutman I, Rehman MU (2015) On spectral radius and energy of complete multipartite graphs. Ars Math Contemp 9:109–113

    Article  MathSciNet  Google Scholar 

  • Sun S, Das KC (2020) Normalized Laplacian eigenvalues with chromatic number and independence number of graphs. Linear Multilinear Algebra 68(1):63–80

    Article  MathSciNet  Google Scholar 

  • Xu K, Das KC, Zhang XD (2016) Ordering connected graphs by their Kirchhoff indices. Int J Comput Math 93(10):1741–1755

    Article  MathSciNet  Google Scholar 

  • Yarahmadi Z (2016) Study of the bipartite edge frustration of graphs. In: Ashrafi A, Diudea M (eds) Distance, symmetry, and topology in carbon nanomaterials. Springer, New York

    Google Scholar 

  • Zhang XD (2004) On the Laplacian spectra of graphs. ARS Comb 72:191–198

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors thank the two anonymous referees for their valuable comments which have considerably improved the presentation of this paper. The first author is supported by National Natural Science Foundation of China (Grant Nos. 11901525 and 11801512) and Zhejiang Provincial Natural Science Foundation of China (LY20A010005). The second author is supported by the National Research Foundation of the Korean government with grant No. 2017R1D1A1B03028642.

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Authors and Affiliations

  1. School of Science, Zhejiang University of Science and Technology, Hangzhou, 310023, Zhejiang, People’s Republic of China

    Shaowei Sun

  2. Department of Mathematics, Sungkyunkwan University, Suwon, 16419, Republic of Korea

    Kinkar Chandra Das

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  1. Shaowei Sun
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  2. Kinkar Chandra Das
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Correspondence to Kinkar Chandra Das.

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Communicated by Carlos Hoppen.

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Sun, S., Das, K.C. On the normalized Laplacian spectral radii of a graph and its line graph. Comp. Appl. Math. 39, 283 (2020). https://doi.org/10.1007/s40314-020-01340-2

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  • DOI: https://doi.org/10.1007/s40314-020-01340-2

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