A Cayley graph Γ = Cay(G,S) is said to be normal for a finite group G, if the right regular representation R(G) is normal in the full automorphism group Aut(Γ) of Γ. In this paper we investigate the normality of Cayley graphs of groups of order a product of two distinct primes, by determining all nonnormal Cayley graphs of these groups.

The paper describes a construction of abstract polytopes from Cayley graphs of symmetric groups. Given any connected graph G with p vertices and q edges, we associate with G a Cayley graph G(G) of the symmetric group Sp and then construct a vertex-transitive simple polytope of rank q, the graphicahedron, whose 1-skeleton (edge graph) is G(G). The graphicahedron of a graph G is a generalization ...

Sense of direction is a property of the labelling of (possibly anonymous) networks which allows to assign coherently local identifiers to other processors on the basis of the route followed by incoming messages. A graph has minimal sense of direction whenever it has sense of direction and the number of colours equals its maximum outdegree. We prove that an outregular digraph with minimal weak s...

Let G be a vertex colored graph. The minimum number χ(G) of colors needed for coloring of a graph G is called the chromatic number. Recently, Adiga et al. [1] have introduced the concept of color energy of a graph Ec(G) and computed the color energy of few families of graphs with χ(G) colors. In this paper we derive explicit formulas for the color energies of the unitary Cayley graph Xn, the co...

In this paper, we focus on the design of network topology to achieve fast information distribution. We present the information distribution performance of Borel Cayley graphs, a family of pseudo-random graphs, is far superior than that of other well-known graph families. To demonstrate the effectiveness of this pseudo-random approach, we compare the convergence speed of the average consensus pr...

Let Γ = Cay(G, S) and G ≤ X ≤ AutΓ. We say Γ is (X, 1)-regular Cayley graph if X acts regularly on its arcs. Γ is said to be corefree if G is core-free in some X ≤ Aut(Cay(G, S)). In this paper, we prove that if an (X, 1)-regular Cayley graph of valency 5 is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free ...

Hexagonal mesh and torus, as well as honeycomb and certain other pruned torus networks, are known to belong to the class of Cayley graphs which are node-symmetric and possess other interesting mathematical properties. In this paper, we use Cayley-graph formulations for the aforementioned networks, along with some of our previous results on subgraphs and coset graphs, to draw conclusions relatin...

Let G be a finite group, and let 1G 6∈ S ⊆ G. A Cayley di-graph Γ = Cay(G,S) of G relative to S is a di-graph with a vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if yx−1 ∈ S. Further, if S = S−1 := {s−1|s ∈ S}, then Γ is undirected. Γ is conected if and only if G = 〈s〉. A Cayley (di)graph Γ = Cay(G,S) is called normal if the right regular representation of G is a ...

The connective constant μ(G) of an infinite transitive graph G is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the th...