arXiv:1510.03605v1 [math.CO] 13 Oct 2015 RELATIVE CAYLEY GRAPHS OF FINITE GROUPS M. FARROKHI D. G., M. RAJABIAN, AND A. ERFANIAN Abstract. The relative Cayley graph of a group G with respect to its proper subgroup H, is a graph whose vertices are elements of G and two vertices h ∈ H and g ∈ G are adjacent if g = hc for some c ∈ C, where C is an inversed-closed subset of G. We study the relative Cayley graphs and, among other results, we discuss on their connectivity and forbidden structures, and compute some of their important numerical invariants. 1. Introduction Cayley graphs was introduced by Arthur Cayley [4] in 1878 to give a geometrical representation of groups by means of a set of generators. This translates groups into geometrical objects which can be studied form the geometrical view. In particular, it provides a rich source of highly symmetric graphs, known as transitive graphs, which plays a central role in many graph theoretical problems as well as group theo- retical problems, like expanders, width of groups, representation of interconnection networks, Hamiltonian paths and cycles that naturally arise in computer science and etc. We intent to introduce and study special subgraphs of the Cayley graphs of a group G with respect to a given proper subgroup H of G, called the relative Cayley graphs. The relative Cayley graph of G with respect to H , denoted by Γ = Cay(G,H,C), is a graph whose vertices are elements of G such that two vertices x and y are adjacent if x or y belongs to H and x −1 y ∈ C for some inversed closed subset C of G \{1}. Clearly, Γ has an induce subgraph Γ ′ = Cay(H,H ∩ C), which is itself a Cayley graph. Relative Cayley graphs with respect to specific subgroups of a group G provides a good source of subgraphs in covering the whole Cayley graph of G. Also, as we shall see in section 3, they give a criterion for a group to be an ABA-group. In this paper, we will investigates some combinatorial and structural properties of relative Cayley graphs. In section 2, we shall obtain preliminary results on valencies and regularity of relative Cayley graphs. In section 3, we pay attention to the connectivity and diameter problems on relative Cayley graphs. We give necessary and sufficient conditions for a relative Cayley to be connected and obtain sharp upper bounds for its diameter. In section 4, the numerical invariants of relative Cayley graphs will be considered. We will determine the explicit value of independence number, dominating number, edge independence number and edge covering number as well as edge chromatic number of relative Cayley graphs. Also, we obtain lower and upper bounds for the clique number and an upper bound for 2000 Mathematics Subject Classification. Primary 05C25, 05C40; Secondary 05C07, 05C69, 05C15. Key words and phrases. Relative Cayley graph, Cayley graph, connectivity, numerical invari- ants, forbidden structures, ABA-group. 1