The geometry of the Cayley graphs of monoids defined by regular confluent monadic rewriting systems is studied. Using geometric and combinatorial arguments, these Cayley graphs are proved to be hyperbolic, and the monoids to be word-hyperbolic in the Duncan–Gilman sense. The hyperbolic boundary of the Cayley graph is described in the case of finite confluent monadic rewriting systems.

A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are ba...

We compare two load balancing techniques for Cayley graphs based on information and load exchange between neighboring vertices. In the rst scheme, called natural di usion, each vertex gives (or receives) a xed part of the load di erence to (from) its direct neighbors. In the second scheme, called Cayley di usion, each vertex successively gives (or receives) a part of the load di erence to (or f...

Let Γ be an additive group. For S ⊆ Γ, 0 6∈ S and S = {−s : s ∈ S} = S, the Cayley graph G = C(Γ, S) is the undirected graph having vertex set V (G) = Γ and edge set E(G) = {(a, b) : a − b ∈ S}. The Cayley graph G = C(Γ, S) is regular of degree |S|. For a positive integer n > 1, the unitary Cayley graph Xn = C(Zn,Z∗n) is defined by the additive group of the ring Zn of integers modulo n and the ...

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...