‎let $s$ be a subset of a finite group $g$‎. ‎the bi-cayley graph ${rm bcay}(g,s)$ of $g$ with respect to $s$ is an undirected graph with vertex set $gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid xin g‎, ‎ sin s}$‎. ‎a bi-cayley graph ${rm bcay}(g,s)$ is called a bci-graph if for any bi-cayley graph ${rm bcay}(g,t)$‎, ‎whenever ${rm bcay}(g,s)cong {rm bcay}(g,t)$ we have $t=gs^alpha$ for some $...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

For a group T and a subset S of T , the bi-Cayley graph BCay(T, S) of T with respect to S is the bipartite graph with vertex set T×{0, 1} and edge set {{(g, 0), (sg, 1)} | g ∈ T, s ∈ S}. In this paper, we investigate cubic bi-Cayley graphs of finite nonabelian simple groups. We give several sufficient or necessary conditions for a bi-Cayley graph to be semisymmetric, and construct several infin...

For a finite group G and a subset S ⊆ G (possibly, S contains the identity of G), the bi-Cayley graph BCay(G, S) of G with respect to S is the graph with vertex set G × {0, 1} and with edge set {(x, 0), (sx, 1)|x ∈ G, s ∈ S}. 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) ∼= BCay(G, T ) we have T = gS , for some g ∈ G, α ∈ Aut(G)....

The bi-Cayley graph of a finite group G with respect to a subset S ⊆ G, which is denoted by BCay(G,S), is the graph with vertex set G× {1, 2} and edge set {{(x, 1), (sx, 2)} | x ∈ G, s ∈ S}. A finite group G is called a 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 a...

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, we give a characterization of cubic nonCayley vertex-transitive bi-Cayley graphs over a regular p-group, where p > 5 is a prime. As an application, a classification of cubic non-Cayley vertex-transitive graphs of order 2p3 is given for each prime p.

The purpose of this paper is the study of Cayley graph associated to a semihypergroup(or hypergroup). In this regards first  we associate a Cayley graph to every semihypergroup and then we study theproperties of this graph, such as  Hamiltonian cycles in this  graph.  Also, by some of examples we will illustrate  the properties and behavior of  these Cayley  graphs, in particulars we show that ...

In this paper, the composite order Cayley graph Cay(G, S) is introduced, where G is a group and S is the set of all composite order elements of G. Some graph parameters such as diameter, girth, clique number, independence number, vertex chromatic number and domination number are calculated for the composite order Cayley graph of some certain groups. Moreover, the planarity of composite order Ca...

A Cayley (resp. bi-Cayley) graph on a dihedral group is called dihedrant bi-dihedrant). In 2000, classification of trivalent arc-transitive dihedrants was given by Marušič and Pisanski, several years later, non-arc-transitive order 4p or 8p (p prime) were classified Feng et al. As generalization these results, our first result presents dihedrants. Using this, complete vertex-transitive non-Cayl...

The original aim of this paper is to construct a graph associated to a vector space. By inspiration of the classical definition for the Cayley graph related to a group we define Cayley graph of a vector space. The vector space Cayley graph ${rm Cay(mathcal{V},S)}$ is a graph with the vertex set the whole vectors of the vector space $mathcal{V}$ and two vectors $v_1,v_2$ join by an edge whenever...