ON THE EIGENVALUES OF N-CAYLEY GRAPHS: A SURVEY
Abstract
A graph Γ is called an n-Cayley graph over a group G if Aut(Γ) contains a semi-regular subgroup isomorphic to G with n orbits. In this paper, we review some recent results and future directions around the problem of computing the eigenvalues on n-Cayley graphs.
Keywords
Full Text:
PDFReferences
M. Arezoomand and B. Taeri: On the characteristic polynomial of n-Cayley digraphs, Electron. J. Combin., 20(3) (2013), # P57.
M. Arezoomand and B. Taeri: Normality of 2-Cayley digraphs, Discrete Math., 338 (2015), 41-47.
M. Arezoomand and B. Taeri: Isomorphisms of finite semi-Cayley graphs, Acta Math. Sin. Engl. Ser., 31(4) (2015) 715-730.
M. Arezoomand and B. Taeri: A classification of finite groups with integral bi-Cayley graphs, Trans. Combin., 4(4) (2015) 55-61.
M. Arezoomand and B. Taeri: Finite BCI-groups are solvable, Int. J. Group Theory, 5(2) (2016) 1-6.
M. Arezoomand and B. Taeri: Which elements of a finite group are non-vanishing?, Bull. Iranian Math. Soc., 42(5) (2016) 1097-1106.
M. Arezoomand and A. R. Ashrafi: On the spectrum of 2-type bicirculants, Proceeding of 10th Graph Theory and Algebraic Combinatorics Conference, Yazd, Iran, 2017, pp. 10-13.
M. Arezoomand and B. Taeri: Finite groups admitting a connected cubic integral bi-Cayley graph, Alg. Struc. Appl., 5(2) (2018) 35-43.
N. Biggs: Algebraic Graph Theory, Cambridge University Press, 1974.
S.F. Du and D. Marušič:An infinite family of biprimitive semisymmetric graphs, J. Graph Theory 32 (1999) 217-228.
S.F. Du and D. Marušič: Biprimitive graphs of smallest order, J. Algebraic Combin. 9 (1999) 151-156.
X. Gao and Y. Luo: The spectrum of semi-Cayley graphs over abelian groups, Linear Algebra Appl., 432 (2010) 2974-2983.
X. Gao, H. Lü and Y. Hao: The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups, J. Appl. Math. Comput., 51 (1) (2016) 383-395.
F. Harary and A. J. Schwenk: Which graphs have integral spectra? in ”Graphs and Combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C., 1973)”, Lecture Notes in Mathematics 406. Springer, Berlin, 1974, pp. 45-51.
A. Hujdurović, K. Kutnar and D. Marušič: On normality of n-Cayley graphs, Appl. Math. Comput., 332 (2018) 469-476.
G. James and M. Liebeck: Representations and characters of groups, Cambridge University Press, Second Edition, 2001.
I. Kovács, I. Malnič, D. Marušič, and S. Miklavič: One-Matching bi-Cayley graphs over abelian groups, European J. Combin. 30 (2009) 602-616.
I. Kovacs, B. Kuzman, A. Malnič and S. Wilson: Characterization of edge-transitive 4-valent bicirculants, J. Graph Theory, DOI: 10.1002/jgt.20594
K. Kutnar, D. Marušič, S. Miklavič and P. Sparl: Strongly regular tri-Cayley graphs, European J. Combin. 30(4) (2009) 822-832.
Z.P. Lu, C.Q. Wang and M.Y. Xu: Semisymmetric cubic graphs constructed from bi-Cayley graphs of A n , Ars Combin. 80 (2006) 177-187.
X. Liu and S. Zhou: Eigenvalues of Cayley graphs: arXiv: 1809.09829 [math.CO].
K.H. Leung and S.L. Ma: Partial difference triples, J. Algebraic Combin. 2 (1993) 397-409.
L. Martinez: Strongly regular m-Cayley circulant graphs and digraphs: Ars Math. Contemp. 8 (2015) 195-213.
D. Maruˇ siˇ c and D. Marui: Strongly regular bicirculants and tricirculants, Ars Combin. 25 (1988) 11-15.
M. J. D. Resmi and D. Jungnickel: Strongly regular Semi-Cayley graphs, J.Algebraic Combin. 1 (1992), 171-195.
G. Sabidussi: Vertex-transitive graphs, Monatsh. Math., 68 (1964) 426–438.
P. Diaconis and M. Shahshahani: Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete, 57 (1981) 159-179.
J. X Zhou: Every finite group has a normal bi-Cayley graph, Ars Math. Contemp., 14 (2018) 177-186.
H. Zou and J.X. Meng: Some algebraic properties of Bi-Cayley graphs, Acta Math. Sin. (Chin. Ser.) 50 (5) (2007) 1075-1080.
DOI: https://doi.org/10.22190/FUMI1903563A
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)