The automorphism groups Aut(C(G, X)) and Aut(CM(G, X, p)) of a Cayley graph C(G, X) and a Cayley map CM(G, X, p) both contain an isomorphic copy of the underlying group G acting via left translations. In our paper, we show that both automorphism groups are rotary extensions of the group G by the stabilizer subgroup of the vertex 1G . We use this description to derive necessary and sufficient co...

The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every nontrivial distance-regular Cayley graph on a dihedral group...

We generalise the biswapped network Bsw(G) to obtain a multiswapped network Msw(H ;G), built around two graphs G and H . We show that the network Msw(H ;G) lends itself to optoelectronic implementation and examine its topological and algorithmic. We derive the length of a shortest path joining any two vertices in Msw(H ;G) and consequently a formula for the diameter. We show that if G has conne...

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory. Thus he paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes a...

Abstract All Cayley representations of the distant graph $$\Gamma _{\mathbb Z}$$ Γ Z over integers are characterized as Neumann subgroups extended modular group. Possible structures revealed and it is shown that every such structure can be realized.

A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).

For the notion of neighbor-connectivity in graphs whenever a vertex is subverted the entire closed neighborhood of the vertex is deleted from the graph. The minimum number of vertices whose subversion results in an empty, complete, or disconnected subgraph is called the neighborconnectivity of the graph. Gunther, Hartnell, and Nowakowski have shown that for any graph, neighbor-connectivity is b...

We will discuss three ways to bound the chromatic number on a Cayley graph. (a) If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. (b) We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency gre...

Let Γ be a Cayley graph of the permutation group generated by a transposition tree T on n vertices. In an oft-cited paper [2] (see also [14]), it is shown that the diameter of the Cayley graph Γ is bounded as diam(Γ) ≤ max π∈Sn {