A classification of finite groups with integral bi-Cayley graphs

Document Type : Research Paper


1 Departmant of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran

2 Department of Mathematics, Isfahan University of Technology, Isfahan, Iran


The bi-Cayley graph of a finite group $G$ with respect to a subset $S\subseteq G$‎, ‎which is denoted by $BCay(G,S)$‎, ‎is the graph with‎ ‎vertex set $G\times\{1,2\}$ and edge set $\{\{(x,1)‎, ‎(sx,2)\}\mid x\in G‎, ‎\ s\in S\}$‎. ‎A‎ ‎finite group $G$ is called a \textit{bi-Cayley integral group} if for any subset $S$ of‎ ‎$G$‎, ‎$BCay(G,S)$ is a graph with integer eigenvalues‎. ‎In this paper we prove‎ ‎that a finite group $G$ is a bi-Cayley integral group if and only if $G$ is isomorphic to‎ ‎one of the groups $\Bbb Z_2^k$‎, ‎for some $k$‎, ‎$\Bbb Z_3$ or $S_3$‎.


Main Subjects

A. Ab dollahi and E. Vatando ost (2009). Which Cayley graphs are integral?. Electron. J. Combin.. 16 (1), 1-17 A. Ab dollahi and M. Jazaeri (2014). Groups all of whose undirected Cayley graphs are integral. European J. Combin.. 38, 102-109 O. Ahmadi, N. Alon, L. F. Blake and I. E. Shparlinski (2009). Graphs with integral sp ectrum. Linear Algebra Appl.. 430, 547-552 A. Ahmady, J. P. Bell and B. Mohar (2014). Integral Cayley graphs and groups. SIAM J. Discrete Math.. 28 (2), 685-701 M. Arezo omand and B. Taeri (2013). On the characteristic p olynomial of n -Cayley digraphs. Electron. J. Combin.. 20 (3), 1-14 K. Balinska, D. Cvetkovic, Z. Radosavljevic, S. Simic and D. Stevanovic (2002). A survey on integral graphs. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.. 13, 42-65 A. E. Brouwer and W. H. Haemers (2012). Spectra of graphs. Springer, Berlin. X. Gao and Y. Luo (2010). The sp ectrum of semi-Cayley graphs over ab elian groups. Linear Algebra Appl.. 432, 2974-2983 F. Harary and A. J. Schwenk (1974). Which graphs have integral sp ectra?. Graphs and Combinatorics (Pro c. Capital Conf., George Washington Univ., Washington, D. C., 1973), Lecture Notes in Mathematics, Springer, Berlin. 406, 45-51 W. Jin and W. Liu (2010). A classi cation of non-ab elian simple 3-BCI-groups. European J. Combin.. 31, 1257-1264 W. Klotz and T. Sander (2010). Integral Cayley graphs over ab elian groups. Electron. J. Combin.. 17, 1-13 I. Kovacs, A. Malnic, D. marusic and S. Miklavic (2009). One-Matching bi-Cayley graphs over ab elian groups. European J. Combin.. 30, 602-616 M. J. de Resmini and D. Jungnickel (1992). Strongly regular semi-Cayley graphs. J. Algebraic Combin.. 1, 171-195 J. P. Serre (1997). Linear representations of nite groups. Graduate Texts in Mathematics, Springer-Verlag, New York. 42 S. J. Xu, W. Jin, Q. Shi, Y. Zhu and J. J. Li (2008). The BCI-prop erty of the Bi-Cayley graphs. J. Guangxi Norm. Univ.: Nat. Sci. Edition. 26, 33-36
Volume 4, Issue 4 - Serial Number 4
December 2015
Pages 55-61
  • Receive Date: 14 July 2014
  • Revise Date: 22 January 2015
  • Accept Date: 23 January 2015
  • Published Online: 01 December 2015