Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 12, 2017

Distance between the spectra of graphs with respect to normalized Laplacian spectra

  • Mojgan Afkhami , Mehdi Hassankhani and Kazem Khashyarmanesh EMAIL logo

Abstract

Let Gn and Gn be two nonisomorphic graphs on n vertices with spectra (with respect to the adjacency matrix)

λ1λ2λnandλ1λ2λn,

respectively. Define the distance between the spectra of Gn and Gn as

λ(Gn,Gn)=i=1n(λi-λi)2(or use i=1n|λi-λi|).

Define the cospectrality of Gn by

cs(Gn)=min{λ(Gn,Gn):Gn not isomorphic to Gn}.

In this paper, we investigate cs(Gn) for special classes of graphs with respect to normalized Laplacian spectra and we find cs(Kn), cs(nK1) and cs(K2+(n-2)K1) (n2). We also find an upper bound for csn.

MSC 2010: 05C50; 05C31

References

[1] A. Abdollahi, S. Janbaz and M. R. Oboudi, Distance between spectra of graphs, Linear Algebra Appl. 466 (2015), 401–408. 10.1016/j.laa.2014.10.020Search in Google Scholar

[2] A. Abdollahi and M. R. Oboudi, Cospectrality of graphs, Linear Algebra Appl. 451 (2014), 169–181. 10.1016/j.laa.2014.02.052Search in Google Scholar

[3] F. R. K. Chung, Spectral Graph Theory, CBMS Reg. Conf. Ser. Math. 92, American Mathematical Society, Providence, 1997. Search in Google Scholar

[4] C. Godsil and G. Royle, Algebraic Graph Theory, Grad. Texts in Math. 207, Springer, New York, 2001. 10.1007/978-1-4613-0163-9Search in Google Scholar

[5] A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J. 20 (1953), 37–39. 10.1215/S0012-7094-53-02004-3Search in Google Scholar

[6] X. Li and Y. Yang, Sharp bounds for the general Randić index, MATCH Commun. Math. Comput. Chem. 51 (2004), 155–166. Search in Google Scholar

[7] D. Stevanović, Research problems from the Aveiro workshop on graph spectra, Linear Algebra Appl. 423 (2007), no. 1, 172–181. 10.1016/j.laa.2006.11.027Search in Google Scholar

[8] T. Tao, Topics in Random Matrix Theory, Grad. Stud. Math. 132, American Mathematical Society, Providence, 2012. 10.1090/gsm/132Search in Google Scholar

Received: 2015-05-27
Accepted: 2015-09-17
Published Online: 2017-11-12
Published in Print: 2019-06-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2017-0051/html
Scroll to top button