JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2022, Vol. 57 ›› Issue (10): 59-65.doi: 10.6040/j.issn.1671-9352.0.2022.004

Previous Articles    

Ramanujan unitary one-matching bi-Cayley graphs over finite commutative rings

GOU Xiao-li, WANG Wei-zhong*   

  1. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, Gansu, China
  • Published:2022-10-06

PDF (PC)

126

Abstract: Let R be a finite commutative ring with unit element 1≠0, and let GR=BC(R; R×, R×, {0})denote the one-matching bi-Cayley graph over R, where R× is the set of units of R. A k-regular graph G is called a Ramanujan graph if any eigenvalue λ of G with |λ|≠k satisfies |λ|≤2(k-1)1/2. A necessary and sufficient condition for GR and its line graph to be Ramanujan is given.

Key words: unitary one-matching bi-Cayley graph, line graph, local ring, finite commutative ring, Ramanujan graph

CLC Number: 

  • O157.5
[1] CAYLEY A. On the theory of group[J]. Proceedings of the London Mathematical Society, 1878, 9:126-233.
[2] DE RESMINI M J, JUNGNICKEL D. Strongly regular semi-Cayley graphs[J]. Journal of Algebraic Combinatorics, 1992, 1(2):171-195.
[3] KOVÁCS I, MALNIC A, MARUŠIC D, et al. One-matching bi-Cayley graphs over Abelian groups[J]. European Journal of Combinatorics, 2009, 30(2):602-616.
[4] ZHOU J X, FENG Y Q. Cubic bi-Cayley graphs over Abelian groups[J]. European Journal of Combinatorics, 2014, 36:679-693.
[5] ZHOU J X, FENG Y Q. The automorphisms of bi-Cayley graphs[J]. Journal of Combinatorial Theory, Series B, 2016, 116:504-532.
[6] LIU X G. Energies of unitary one-matching bi-Cayley graphs over finite commutative rings[J]. Discrete Applied Mathematics, 2019, 259:170-179.
[7] LUBOTZKY A, PHILLIPS R, SARNAK P. Ramanujan graphs[J]. Combinatorica, 1988, 8(3):261-277.
[8] DAVIDOFF G, SARNAK P, VALETTE A. Elementary number theory, group theory and Ramanujan graphs[M]. Cambridge: Cambridge University Press, 2003.
[9] LIU X G, LI B L. Distance powers of unitary Cayley graphs[J]. Applied Mathematics and Computation, 2016, 289:272-280.
[10] LIU X G, ZHOU S M. Spectral properties of unitary Cayley graphs of finite commutative rings[J]. The Electronic Journal of Combinatorics, 2012, 19(4):1-19.
[11] LIU X G, ZHOU S M. Quadratic unitary Cayley graphs of finite commutative rings [J]. Linear Algebra and its Applications, 2015, 479:73-90.
[12] HOORY S, LINIAL N, WIGDERSON A. Expander graphs and their applications[J]. Bulletin of the American Mathematical Society, 2006, 43(4):439-561.
[13] MURTY M R. Ramanujan graphs[J]. Journal of the Ramanujan Mathematics Society, 2003, 18(1):1-20.
[14] LIU X G, ZHOU S M. Eigenvalues of Cayley graphs[J]. The Electronic Journal of Combinatorics, 2022, 29(2):1-164.
[15] LIU X G, YAN C X. Unitary homogeneous bi-Cayley graphs over finite commutative rings [J]. Journal of Algebra and Its Applications, 2020, 19(9):2050173.
[16] ATIYAH M F, MACDONALD I G. Introduction to commutative algebra[M]. London: Addison-Wesley, 1969.
[17] DUMMIT D S, FOOTE R M. Abstract algebra[M]. 3th ed. New York: Wiley, 2003.
[18] AKHTAR R, BOGGESS M, JACKSON-HENDERSON T, et al. On the unitary Cayley graph of a finite ring[J]. The Electronic Journal of Combinatorics, 2009, 16(1):1-13.
[19] GANESAN N. Properties of rings with a finite number of zero-divisors[J]. Mathematische Annalen, 1964, 157(3):215-218.
[1] LU Peng-li, LIU Wen-zhi. On the generalized distance spectrum of graphs [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(9): 19-28.
[2] WANG Wan-yu, MENG Ji-xiang*, ZHAO Xue-bing. The restricted neighbor connectivity of  line graphs [J]. J4, 2012, 47(2): 56-59.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
No Suggested Reading articles found!
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd