For a finite group G and subset S of G, the Haar graph H(G,S) is a bipartite regular graph, defined as a regular G-cover of a dipole with |S| parallel arcs labelled by elements of S. If G is an abelian group, then H(G,S) is well-known to be a Cayley graph; however, there are examples of non-abelian groups G and subsets S when this is not the case. In this paper we address the problem of classif...

This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and corefree), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various c...

Let S be a set of transpositions that generates the symmetric group Sn, where n ≥ 3. The transposition graph T (S) is defined to be the graph with vertex set {1, . . . , n} and with vertices i and j being adjacent in T (S) whenever (i, j) ∈ S. We prove that if the girth of the transposition graph T (S) is at least 5, then the automorphism group of the Cayley graph Cay(Sn, S) is the semidirect p...

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V,E) and σ : E → {+,−} is a function from the edge set E of Su into the set {+,−}. For a positive integer n > 1, the unitary Cayley graph Xn is the graph whose vertex set is Zn, the integers modulo n and if Un denotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a...

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.