We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter CG for Cayley graphs G that has significant application to percolation. For a minimal cutset of G and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is CG. We show that if it is fi...

We show that every connected graph can be isometrically (i.e., as a distance preserving subgraph) embedded in some connected graph as its median. As an auxiliary result we also show that every connected graph is an isometric subgraph of some Cayley graph.

In the search for regular, undirected dense graphs for interconnection networks, Chudnovsky et al. found certain Cayley graphs that are the densest degree-four graphs known for an interesting range of diameters [l]. However, the group theoretic representation of Cayley graphs makes the development of effective routing algorithms difficult. This paper shows that all finite Cayley graphs can be r...

Cayley graphs and elementary group theory are used to eeciently nd minimum length permutation routes in a bus interconnection network. Cayley graphs have been used extensively to design interconnection networks and provide a natural setting for studying point-to-point routing 1, 2, 3, 4], but the technique has not been widely used for the more important problem of permutation routing. This is d...

Let G be a finite group, and X a subset of G. The Cayley graph of G with respect to X , written Cay(G,X) has two different definitions in the literature. The vertex set of this graph is the group G. In one definition, there is an arc from g to xg for all g ∈ G and x ∈ X ; in the other definition, for the same pairs (g,x), there is an arc from g to gx. Cayley graphs are generalised by coset grap...

Our aim in this note is to present a transitive graph that we conjecture is not quasi-isometric to any Cayley graph. No such graph is currently known. Our graph arises both as an abstract limit in a suitable space of graphs and in a concrete way as a subset of a product of trees.

In this paper we consider the Cayley graph Gf associated to a Boolean function f and we use it to investigate some of the cryptographic properties of f . We derive necessary (but not sufficient) conditions for a Boolean function to be bent. We also find a complete characterization of the propagation characteristics of f using the topology of its associated Cayley graph Gf . Finally, some inequa...

Up until the Bourgain-Gamburd 2005 breakthrough the only known ways to turn SLd(Fp) into an expander graph (i.e. to find a generating set of small size whose associated Cayley graph has a good spectral gap) was either through property (T ) (as in the Margulis construction) when d > 3 or through the Selberg property (and the dictionary between combinatorial expansion of the Cayley graphs and the...

The unipotent subgroup of a finite group of Lie type over a prime field Fp comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n2 log p, where n is the rank of the associated Lie...