Czechoslovak Mathematical Journal

, Volume 68, Issue 1, pp 1–17 | Cite as

Graphs with small diameter determined by their D-spectra

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Abstract

Let G be a connected graph with vertex set V(G) = {v1, v2,..., v n }. The distance matrix D(G) = (d ij )n×n is the matrix indexed by the vertices of G, where d ij denotes the distance between the vertices v i and v j . Suppose that λ1(D) ≥ λ2(D) ≥... ≥ λ n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.

Keywords

distance spectrum distance characteristic polynomial distance characteristic polynomial D-spectrum deter- mined by its D-spectrum 

MSC 2010

05C50 

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References

  1. [1]
    S. M. Cioabă, W. H. Haemers, J. R. Vermette, W. Wong: The graphs with all but two eigenvalues equal to ±1. J. Algebra Comb. 41 (2015), 887–897.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    D. M. Cvetković, M. Doob, H. Sachs: Spectra of Graphs. Theory and Applications. J. A. Barth Verlag, Heidelberg, 1995.MATHGoogle Scholar
  3. [3]
    H. H. Günthard, H. Primas: Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit systemen konjugierter Bindungen. Helv. Chim. Acta 39 (1956), 1645–1653. (In German.) doiCrossRefGoogle Scholar
  4. [4]
    Y.-L. Jin, X.-D. Zhang: Complete multipartite graphs are determined by their distance spectra. Linear Algebra Appl. 448 (2014), 285–291.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    L. Lu, Q. X. Huang, X. Y. Huang: The graphs with exactly two distance eigenvalues different from −1 and −3. J. Algebr. Comb. 45 (2017), 629–647.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    H. Q. Lin: On the least distance eigenvalue and its applications on the distance spread. Discrete Math. 338 (2015), 868–874.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    H. Q. Lin, Y. Hong, J. F. Wang, J. L. Shu: On the distance spectrum of graphs. Linear Algebra Appl. 439 (2013), 1662–1669.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    H. Q. Lin, M. Q. Zhai, S. C. Gong: On graphs with at least three distance eigenvalues less than −1. Linear Algebra Appl. 458 (2014), 548–558.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    R. F. Liu, J. Xue, L. T. Guo: On the second largest distance eigenvalue of a graph. Linear Multilinear Algebra 65 (2017), 1011–1021.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    E. R. van Dam, W. H. Haemers: Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272.Google Scholar
  11. [11]
    E. R. van Dam, W. H. Haemers: Developments on spectral characterizations of graphs. Discrete Math. 309 (2009), 576–586.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Xue, R. F. Liu, H. C. Jia: On the distance spectrum of trees. Filomat 30 (2016), 1559–1565.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityHenanChina

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