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 {

A well known unresolved conjecture states that every Cayley graph on a solvable group G has a 1-factorization. We show that if the commutator subgroup of of such a group is an elementary abelian p-group, then every quartic Cayley graph on G has a 1-factorization.

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...

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...

For a graph G = (V,E), a subset F ⊂ V (G) is called an Rk-vertex-cut of G if G − F is disconnected and each vertex u ∈ V (G) − F has at least k neighbors in G − F . The Rk-vertex-connectivity of G, denoted by κ (G), is the cardinality of the minimum Rk-vertex-cut of G, which is a refined measure for the fault tolerance of network G. In this paper, we study κ2 for Cayley graphs generated by k-tr...

Given a group G, we can construct a graph relating the elements of G to each other, called the Cayley graph. Using Fourier analysis on a group allows us to apply our knowledge of the group to gain insight into problems about various properties of the graph. Ideas from representation theory are powerful tools for analysis of groups and their Cayley graphs, but we start with some simpler theory t...

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...