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.

we characterize the rational subsets of a finite group and discuss the relations to integral cayley graphs.

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are disti...

Let Γ be a finite, additive group, S ⊆ Γ, 0 6∈ S, − S = {−s : s ∈ S} = S. The undirected Cayley graph Cay(Γ, S) has vertex set Γ and edge set {{a, b} : a, b ∈ Γ, a − b ∈ S}. A graph is called integral, if all of its eigenvalues are integers. For an abelian group Γ we show that Cay(Γ, S) is integral, if S belongs to the Boolean algebra B(Γ) generated by the subgroups of Γ. The converse is proven...

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

Let G be a non-trivial group, S ⊆ G \ {1} and S = S−1 := {s−1 | s ∈ S}. The Cayley graph of G denoted by Γ(S : G) is a graph with vertex set G and two vertices a and b are adjacent if ab−1 ∈ S. A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine all connected cubic integral Cayley graphs. We also introduce some infinite families of connected integra...

‎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 $...

Let G be a finite group, S ⊆ G \ {1} be a set such that if a ∈ S, then a−1 ∈ S, where 1 denotes the identity element of G. The undirected Cayley graph Cay(G, S) ofG over the set S is the graphwhose vertex set is G and two vertices a and b are adjacent whenever ab−1 ∈ S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called i...

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 ...