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On the isomorphisms of bi-Cayley graphs

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Abstract

Let $G$ be a group and $S$ be a subset of $G$. Then $BCay(G,S)$, bi-Cayley graph of $G$ with respect to $S$, is an undirected graph with vertex-set $G\times{1,2}$ and edge-set ${{(g,1),(sg,2)}\mid g\in G, s\in S}$. For $\sigma\in Aut(G)$ and $g\in G$, we have $BCay(G,S)\cong BCay(G,gS^\sigma)$. A bi-Cayley graph $BCay(G,S)$ is called a $BCI$-graph if for any bi-Cayley graph $BCay(G,T)$, whenever $BCay(G,S)\cong BCay(G,T)$ we have $T=gS^\sigma$ for some $g\in G$ and $\sigma\in Aut(G)$. A group $G$ is called a $BCI$-group if every bi-Cayley graph of $G$ is a $BCI$-graph. In this lecture, we discuss recent results and future directions of classifying finite $BCI$-groups.

Introduction and results

A fundamental problem about Cayley graphs is the so called isomorphism problem, that is, given two Cayley graphs $Cay(G,S)$ and $Cay(H,T)$ determine whether or not $Cay(G,S)\cong Cay(H,T)$. It follows quickly from the definition that for any automorphism $\alpha\in Aut(G)$, the graphs $Cay(G,S)$ and $Cay(G,S^\alpha)$ are isomorphic, namely, $\alpha$ induces an isomorphism between these graphs. Such an isomorphism is also called a Cayley isomorphism.

In 1967, Adam [1] conjectured that two Cayley graphs over the cyclic group are isomorphic if and only if there is a Cayley isomorphism which maps one to the other. Soon afterwards, Elspas and Turner [4] found the counterexample for $n=8$. This also motivated the following definition. A Cayley graph $Cay(G,S)$ has the $CI$-property (for short, it is a $CI$-graph) if for any Cayley graph $Cay(G,T), Cay(G,S)\cong Cay(G,T)$ implies that $T = S^\alpha$ for some $\alpha\in Aut(G)$. Finite $CI$-groups have attracted considerable attention over the last 50 years. The problem of classifying finite $CI$-groups is still open.

In 2008, motivated by the concepts $CI$-graph, $m-BCI$-group and $CI$-group, Xu et al. [13] introduced the concepts $BCI$-graph, $m-BCI$-group and $BCI$-group, respectively. We say that a bi-Cayley graph $BCay(G,S)$ is a $BCI$-graph if whenever $BCay(G,S)\cong BCay(G,T)$ for some subset $T$ of $G,$ the set $T = gS^\alpha$ for some $g\in G$ and automorphism $\alpha\in Aut(G)$. The group $G$ is an $m-BCI$-group if every bi-Cayley graph over $G$ of valency at most $m$ is a $BCI$-graph, and $G$ is a $BCI$-group if every bi-Cayley graph over $G$ is a $BCI$-graph. The theory of $BCI$-graphs and $BCI$-groups is less developed as in the case of $CI$-graphs and $CI$-groups. Several basic properties have been obtained by Jin and Liu in a series of papers [5-8], also by Koike et.al. [9-12] and very recently, by the present author [2-3]. We will discuss the relation between $BCI$-groups and $CI$-groups. In fact, our primary motivation by studying $BCI$-graphs and $BCI$-groups is that these objects can bring new insight into the old problem of classifying $CI$-groups. In [2] it is conjectured that every $BCI$-group is a $CI$-group and it is proved that every group of prime order is a $BCI$-group and every Sylow subgroup of a $BCI$-group is elementary abelian. Also, in[3], it is proved that every $BCI$-group is solvable.

Since every bi-Cayley graph over an abelian group is a Cayley graph over a generalized dihedral group, it seems that classifying finite abelian $BCI$-groups can help to classifying generalized dihedral $CI$-groups. In this lecture, we present some new results about classifying finite abelian $BCI$-groups.

This is a joint work with Majid Arezoomandb.

References:

[1] A. Adam. Research problems 2-10. J. Combin. Theory, 2, 393, 1967.

[2] M. Arezoomand and B. Taeri. Isomorphisms of finite semi-Cayley graphs. Acta Math. Sin. (Engl. Ser.), 31(4) 715-730, 2015.

[3] M. Arezoomand and B. Taeri. Finite BCI-groups are solvable. Int. J. Group Theory, 5(2), 1-6, 2016.

[4] B. Elspas and J. Turner. Graphs with circulants adjacency matrices. J. Combin. Theory, 9, 29-307, 1970.

[5]W. Jin and W. Liu. Two results on BCI-subset of finite groups. Ars Combin., 93,169-173, 2009.

[6]W. Jin and W. Liu, A classification of nonabelian simple 3-BCI-groups. European Journal of Combinatorics, 31 (2010) 1257-1264.

[7] W. Jin and W. Liu. On sylow subgroups of BCI-groups. Util. Math., 86,313-320, 2011.

[8] W. Jin and W. Liu. On isomorphisms of small order bi-Cayley graphs. Util. Math., 92, 317-327, 2013.

[9] H. Koike and I. Kovacs. Arc-transitive cubic abelian bi-cayley graphs and BCI-graphs. FILOMAT, 30(2) (2016) 321-331.

[10] H. Koike and I. Kovacs. A classification of nilpotent 3-BCI groups. submitted.

[11] H. Koike and I. Kovacs. Isomorphic tetravalent circulant Haar graphs. Ars Math. Contemp. 7(2), 215-235, 2014.

[12] H. Koike, I. Kovacs, and T. Pisanski. The number of cyclic configurations of type $(v_3)$ and the isomorphism problem. J. Combin. Des., 22(5),216-229, 2014.

[13] S. J. Xu, W. Jin, Q. Shi, Y. Zhu and J. J. Li, The BCI-property of the Bi-Cayley graphs. J. Guangxi Norm. Univ.: Nat. Sci. Edition 26 (2008) 33-36.