Many large graphs can be constructed from existing smaller graphs by using graph operations, for example, the Cartesian product and the lexicographic product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the lexicographic products of vertextransitive and of edge-transitive graphs. In particula...

Large wireless sensor networks can contain hundreds or thousands of sensor nodes. Due to wireless sensor network’s properties of low-energy-efficiency, large-scale, low cost, and lossy nature, the development of efficient routing protocols for these large and dense wireless sensor networks is an interesting research topic. This research focuses on the design and implementation of protocols for ...

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...

If K = Goφ Z where φ is a tame automorphism of the 1-relator group G, then the combinatorial area of loops in a Cayley graph of G is undistorted in a Cayley graph of K. Examples of distortion of area in fibres of fibrations over the circle are given and a notion of exponent of area distortion is introduced and studied. The inclusion of a finitely generated abelian subgroup in the fundamental gr...

It is a difficult problem in general to decide whether a Cayley graph Cay(G; S) is connected where G is an arbitrary finite group and S a subset of G. For example, testing primitivity of an element in a finite field is a special case of this problem but notoriously hard. In this paper, it is shown that if a Cayley graph Cay(G; S) is known to be connected then its fault tolerance can be determin...

In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph G(10, 2), occurring as the...

We examine a number of countable homogeneous relational structures with the aim of deciding which countable groups can act regularly on them. Since a group X acts regularly on a graph G if and only if G is a Cayley graph for X, we will extend the terminology and say that M is a Cayley object for X if X acts regularly on M. We consider, among other things, graphs, hypergraphs, metric spaces and ...

Given a group or semi-group Γ generated (in the sense of semi-groups) by some subset S ⊂ Γ, the Cayley graph of Γ with respect to S is the oriented graph having vertices γ ∈ Γ and oriented edges (γ, γs)s∈S . In this paper, we consider generating sets S which are not necessarily finite thus yielding oriented Cayley graphs which are perhaps not locally finite. The above situation gives rise to a ...

The adjacency spectrum Spec(Γ) of a graph Γ is the multiset of eigenvalues of its adjacency matrix. Two graphs with the same spectrum are called cospectral. A graph Γ is “determined by its spectrum” (DS for short) if every graph cospectral to it is in fact isomorphic to it. A group is DS if all of its Cayley graphs are DS. A group G is Cay-DS if every two cospectral Cayley graphs of G are isomo...