Eigenvalues of Antiadjacency Matrix of Cayley Graph of Z_n

Juan Daniel, Kiki Ariyanti Sugeng, Nora Hariadi

Abstract


In this paper, we give a relation between the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) and the eigenvalues of the antiadjacency matrix of Cay(Z_n, (Z_n−{0})−S), as well as the eigenvalues of the adjacency matrix of Cay(Z_n, S). Then, we give the characterization of connection set S where the eigenvalues of the antiadjacency matrix of Cay(Z_n, S) are all integers.


Keywords


Antiadjacency matrix, Cayley graph, group Z_n, eigenvalues, adjacency matrix, circulant matrix

Full Text:

Remote PDF

DOI: http://dx.doi.org/10.19184/ijc.2022.6.1.5

References

A. Abdollahi and E. Vatandoost, Which Cayley graphs are integral? Electron. J. Combin. 16(1) (2009), Paper #R122, 17.

B. Alspach, Isomorphism and Cayley graphs on abelian groups, In: G. Hahn and G. Sabidussi, Graph Symmetry, NATO ASI Series (Series C: Mathematical and Physical Sciences), vol 497, Springer, Dordrecht, (1997).

R. Bapat, Graphs and Matrices, Springer, London (2010).

B. F. Chen, E. Ghorbani, and K. B. Wong, On the eigenvalues of certain Cayley graphs and arrangement graphs, Linear Algebra Appl. 444 (2014), 246–253.

F. Harary and A. Schwenk, Which graphs have integral spectra?, Lect. Notes Math., Springer Verlag 406 (1974), 45–50.

K. Kalpakis and Y. Yesha, On the bisection width of the transposition network, Networks 29(1) (1997), 69–76.

W. Klotz and T. Sander, Some properties of unitary Cayley graphs, Electron. J. Combin. 14(1) (2007), Paper #R45, 12.

M. Krebs and A. Shaheen, Expander Families and Cayley Graphs: A Beginner’s Guide, Oxford University Press, New York (2011).

X. Liu and S. Zhou, Eigenvalues of Cayley graphs, arXiv: Combinatorics (2008), 1-115.

L. Lu, Q. Huang, and X. Huang, Integral cayley graphs over dihedral groups, Journal of Algebraic Combinatorics 47(4) (2018), 585–601.

S. M. Mirafzal and A. Zafari, On the spectrum of a class of distance-transitive graphs, Electronic Journal of Graph Theory and Applications 5(1) (2017), 63-69.

Murni, A. E. Hadi, I. Febry, and Abdussakir, Anti-adjacency and Laplacian spectra of inverse graph of group of integers modulo n, IOP Conf. Ser.: Mater. Sci. Eng. 807 (2020) 012033 1-9.

R. P. Oktradifa, S. Aminah, and K. A. Sugeng, Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph, J. Phys.: Conf. Ser. 1722 (2021) 012056 1-9.

W. So, Integral circulant graphs, Discrete Math. 306(1) (2005), 153–158.

R. Stin, S. Aminah, and S. Utama, Characteristic polynomial and eigenvalues of the antiadjacency matrix of cyclic directed prism graph, AIP Conference Proceedings 2168, (2019) 020052 1-8.

V. Vilfred, On circulant graphs, In: R. Balakrishnan, G. Sethuraman, and R.J. Wilson, Graph Theory and its Applications, Narosa Publishing House, Chennai, (2004)


Refbacks

  • There are currently no refbacks.


ISSN: 2541-2205

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

View IJC Stats