Let K be a general nite commutative ring. We refer to a family gn, n = 1, 2, . . . of bijective polynomial multivariate maps of K as a family with invertible decomposition gn = g ng 2 n . . . g k n, such that the knowledge of the composition of g n allows computation of g i n for O(n ) (s > 0) elementary steps. A polynomial map g is stable if all non-identical elements of kind g, t > 0 are of t...

Transitive signatures allow a signer to authenticate edges in a graph in such a way that anyone, given the public key and two signatures on adjacent edges (i, j) and (j, k), can compute a third signature on edge (i, k). A number of schemes have been proposed for undirected graphs, but the case of directed graphs remains an open problem. At CT-RSA 2007, Yi presented a scheme for directed trees b...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

Short cycles connectivity is a generalization of ordinary connectivity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is...

Short cycle connectivity is a generalization of ordinary connectiv-ity. Instead by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two adjacent cycles have at least one common vertex. If all adjacent cycles in the sequence share at least one edge, we talk about edge short cycles connectivity. It is shown that the short cycles connectivity is...

These lecture notes are on automorphism groups of Cayley graphs and their applications to optimal fault-tolerance of some interconnection networks. We first give an introduction to automorphisms of graphs and an introduction to Cayley graphs. We then discuss automorphism groups of Cayley graphs. We prove that the vertex-connectivity of edge-transitive graphs is maximum possible. We investigate ...

A subgroup of the automorphism group a graph Γ is said to be half-arc-transitive on if its action transitive vertex set and edge but not arc Γ. Tetravalent graphs girths 3 4 admitting automorphisms have previously been characterized. In this paper we study examples girth 5. We show that, with two exceptions, all such only directed 5-cycles respect corresponding induced orientation edges. Moreov...