Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

THE SPECTRAL DETERMINATIONS OF THE JOIN OF TWO FRIENDSHIP GRAPHS

  • Received : 2018.01.09
  • Accepted : 2019.02.25
  • Published : 2019.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

The main aim of this study is to characterize new classes of multicone graphs which are determined by their adjacency spectra, their Laplacian spectra, their complement with respect to signless Laplacian spectra and their complement with respect to their adjacency spectra. A multicone graph is defined to be the join of a clique and a regular graph. If n is a positive integer, a friendship graph $F_n$ consists of n edge-disjoint triangles that all of them meet in one vertex. It is proved that any connected graph cospectral to a multicone graph $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ is determined by its adjacency spectra as well as its Laplacian spectra. In addition, we show that if $n{\neq}2$, the complement of these graphs are determined by their adjacency spectra. At the end of the paper, it is proved that multicone graphs $F_n{\nabla}F_n=K_2{\nabla}nK_2{\nabla}nK_2$ are determined by their signless Laplacian spectra and also we prove that any graph cospectral to one of multicone graphs $F_n{\nabla}F_n$ is perfect.

Keywords

HNSHCY_2019_v41n1_67_f0001.png 이미지

Figure 1. Friendship graphs F1, F2, F3 and Fn.

References

  1. A.Z. Abdian and S.M. Mirafzal, On new classes of multicone graphs determined by their spectrums, Alg. Struc. Appl, 2 (2015), 23-34.
  2. A.Z. Abdian, Graphs which are determined by their spectrum, Konuralp. J. Math, 4 (2016), 34-41.
  3. A.Z. Abdian, Two classes of multicone graphs determined by their spectra, J. Math. Ext., 10 (2016), 111-121.
  4. A.Z. Abdian, Graphs cospectral with multicone graphs $K_w$ $\nabla$L(P), TWMS. J. App. and Eng. Math., 7 (2017), 181-187.
  5. A.Z. Abdian, The spectral determinations of the multicone graphs $K_w$ $\nabla$P, arXiv:1706.02661.
  6. A.Z. Abdian and S. M. Mirafzal, The spectral characterizations of the connected multicone graphs $K_w$ $\nabla$LHS and $K_w$ $\nabla$LGQ(3; 9), Discrete Math. Algorithm and Appl (DMAA), 10 (2018), 1850019. https://doi.org/10.1142/S1793830918500192
  7. A.Z. Abdian and S. M. Mirafzal, The spectral determinations of the connected multicone graphs $K_w$ $\nabla$$mP_{17}$ and $K_w$ $\nabla$mS, Czech. Math. J., (2018), 1-14, DOI 10.21136/CMJ.2018.0098-17.
  8. A.Z. Abdian and et al., On the spectral determinations of the connected multicone graphs $K_r{\nabla}sKt$, AKCE Int. J. Graphs and Combin., 10.1016/j.akcej.2018.11.002.
  9. A.Z. Abdian, A. Behmaram and G.H. Fath-Tabar, Graphs determined by signless Laplacian spectra, AKCE Int. J. Graphs and Combin., https://doi.org/10.1016/j.akcej.2018.06.00.
  10. A.Z. Abdian, G.H. Fath-Tabar, and M.R. Moghaddam, The spectral determination of the multicone graphs $K_w{\nabla}C$ with respect to their signless Laplacian spectra, Journal of Algebraic Systems, (to appear).
  11. A.Z. Abdian, Sara Pouyandeh and Bahman Askari, Which multicone graphs $K_m{\nabla}K_n$ are determined by their signless Laplacian spectra? (the proof of a conjecture), Journal of Discrete Mathematical Sciences and Cryptography, (to appear).
  12. R. B. Bapat, Graphs and Matrices, Springer-Verlag, New York, 2010.
  13. N. L. Biggs, Algebraic Graph Theory, Cambridge University press, Cambridge, 1933.
  14. A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Universitext. Springer, New York, 2012.
  15. R. Boulet and B. Jouve, The lollipop graph is determined by its spectrum, Electron. J. Combin., 15 (2008) R74. https://doi.org/10.37236/798
  16. A; Brandstadt, V. B. Leand J. P. Spinrad, Graph classes: a survey, SIAM Monographs on Discrete Math. Appl., 1999.
  17. C. Bu, and J. Zhou, Signless Laplacian spectral characterization of the cones over some regular graphs, Linear Algebra Appl., 9 (2012), 3634-3641.
  18. X. M. Cheng, G. R. W. Greaves, J. H. Koolen, Graphs with three eigenvalues and second largest eigenvalue at most 1, http://de.arxiv.org/abs/1506.02435v1.
  19. S. M. Cioaba, W. H. Haemers, J. R. Vermette and W. Wong, The graphs with all but two eigenvalues equal to${\pm}1$, J. Algebr. Combin, 41 (2013), 887-897. https://doi.org/10.1007/s10801-014-0557-y
  20. D. Cvetkovic, P. Rowlinson and S. Simic, An Introduction to the Theory of graph spectra, London Mathematical Society Student Teyts, 75, Cambridge University Press, Cambridge, 2010.
  21. D. Cvetkovic, P. Rowlinson and S. Simic, Signless Laplacians of finite graphs, Linear Algebra appl., 423.1 (2007): 155-171. https://doi.org/10.1016/j.laa.2007.01.009
  22. M. Doob and W. H. Haemers, The complement of the path is determined by its spectrum, Linear Algebra Appl., 356, (2002) 57-65. https://doi.org/10.1016/S0024-3795(02)00323-3
  23. W. H. Haemers, X. G. Liu and Y. P. Zhang, Spectral characterizations of lollipop graphs, Linear Algebra Appl., 428 (2008), 2415-2423. https://doi.org/10.1016/j.laa.2007.10.018
  24. Y. Hong, J. Shu and K. Fang, A sharp upper bound of the spectral radius of graphs, J. Combin., Theory Ser. B, 81 (2001) 177-183. https://doi.org/10.1006/jctb.2000.1997
  25. HS. H. Gunthard and H. Primas, Zusammenhang von Graph theory und Mo-Theorie von Molekeln mit Systemen konjugierter Bindungen, Helv. Chim. Acta, 39 (1925), 1645-1653. https://doi.org/10.1002/hlca.19560390623
  26. U. Knauer, Algebraic Graph Theory, Morphism, Monoids and Matrices, de Gruyters Studies in Mathematics, Vol. 41, Walter de Gruyters and Co., Berlin and Boston, 2011.
  27. Y. Liu and Y. Q. Sun, On the second Laplacian spectral moment of a graph, Czech. Math. J., 2 (2010), 401-410.
  28. X. Liu and L. Pengli, Signless Laplacian spectral characterization of some joins, Electron. J. Linear Algebra 30.1 (2015), 30.
  29. S.M. Mirafzal and A.Z. Abdian, Spectral characterization of new classes of multicone graphs, Stud. Univ. Babes-Bolyai Math., 62 (3), (2017), 275-286. https://doi.org/10.24193/subbmath.2017.3.01
  30. S.M. Mirafzal and A.Z. Abdian, The spectral determinations of some classes of multicone graphs, Journal of Discrete Mathematical Sciences and Cryptography, 21 (1), (2018), 179-189, DOI: 10.1080/09720529.2017.1379232.
  31. R. Sharafdini and A.Z. Abdian, Signless Laplacian determinations of some graphs with independent edges, Carpathian Math. Publ., 10 (1), (2018), 185-196. https://doi.org/10.15330/cmp.10.1.185-196
  32. W. Peisert, All self-complementary symmetric graph, J. Algebra 240 (2001), 209-229. https://doi.org/10.1006/jabr.2000.8714
  33. P. Rowlinson, The main eigenvalues of a graph: a survey, Appl. Anal. Discrete Math., 1 (2007) 445-471.
  34. E. R. Van Dam and W. H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra. Appl., 373 (2003), 241-272. https://doi.org/10.1016/S0024-3795(03)00483-X
  35. E. R. Van Dam and W. H. Haemers, Developments on spectral characterizations of graphs, Discrete Math., 309 (2009), 576-586. https://doi.org/10.1016/j.disc.2008.08.019
  36. E.R. Van Dam, Nonregular graphs with three eigenvalues, J. Combin.Theory, Series B 73.2 (1998), 101-118. https://doi.org/10.1006/jctb.1998.1815
  37. W. Yi, F. Yizheng, and T. Yingying, On graphs with three distinct Laplacian eigenvalues, Appl. Math., A Journal of Chinese Universities, 22 (2007),478-484. https://doi.org/10.1007/s11766-007-0414-z
  38. J. Wang, H. Zhao, and Q. Huang, Spectral charactrization of multicone graphs, Czech. Math. J., 62 (2012) 117-126. https://doi.org/10.1007/s10587-012-0021-x
  39. W. Wang and C. x. Xu, A sufficient condition for a family of graphs being determined by their generalized spectra, European J. Combin., 27 (2006), 826-840. https://doi.org/10.1016/j.ejc.2005.05.004
  40. J.Wang, F. Belardo, Q. Huang, B. Borovicanin, On the two largest Q-eigenvalues of graphs, Discrete Math., 310 (2010), 2858-2866. https://doi.org/10.1016/j.disc.2010.06.030
  41. J. Wang and Q. Huang, Spectral characterization of generalized cocktail-party graphs, J. Math. Res. Appl., 32 (2012), 666-672.
  42. D. B. West, Introduction to Graph Theory, Upper Saddle River: Prentice hall, 2001.