Anna S. Lladó

The partial line digraph technique in the design of large interconnection networks

The following problem arises in the design of some interconnection networks for distributed systems. Namely, to construct digraphs with given maximum out-degree, reduced diameter, easy routing, good connectivity, and good expandability. To this end, a method based on the concept of partial line digraph is presented. This proposal, which turns out to be a generalization of the so-called line digraph technique, allows digraphs that satisfy all the above-mentioned requirements to be obtained. In particular, it is shown that the partial line digraphs of Kautz digraphs solve the (d, N) digraph problem, i.e. to minimize the diameter D in a digraph of maximum out-degree d and number of vertices N, for any N in the range d/sup D-1/+d/sup D-2/+. . .+1 >

Vosperian and superconnected Abelian Cayley digraphs

A digraphX is said to be Vosperian if any fragment has cardinality either 1 or|V(X)| − d+(X) − 1.A digraph is said to be superconnected if every minimum cutset is the set of vertices adjacent from or to some vertex.In this paper we characterize Vosperian and superconnected Abelian Cayley directed graphs. Our main tool is a difficult theorem of J.H. Kemperman from Additive Group Theory.In particular we characterize Vosperian and superconnected loops network (also called circulants).

On Subsets with Small Product in Torsion-Free Groups

G be a nonabelian torsion-free group. Let C be a finite generating subset of G such that . We prove that, for all subsets B of G with , we have .In particular, a finite subset X with cardinality satisfies the inequality if and only if there are elements , such that the following two conditions hold:(i) .(ii) where .

The connectivity of hierarchical Cayley digraphs

Abstract An ordered generating set of a group is hierarchical when the group generated by the first k generators is a proper subgroup of the group generated by the first k + 1 for each k . Hierarchical Cayley graphs were introduced by Akers and Krishnamurthy in [1] and other related papers as an interesting model to build symmetrical networks. In this paper we study the connectivity of hierarchical Cayley digraphs, and we show that they have maximum connectivity except in a special case. An example of this exceptional case is given. The result generalizes similar statements of Godsil [3], Akers [1] and Hamidoune [5].

Edge-decompositions ofKn,ninto isomorphic copies of a given tree: DECOMPOSINGKn,nINTO TREES

We study the Häggkvist conjecture which states that, for each tree T with n edges, there is an edge-partition of the complete bipartite graph Kn;n into n isomorphic copies of T . We use the concept of bigraceful labelings, introduced in [7], which give rise to cyclic decompositions of Kn;n. When a tree T of size n is not known to be bigraceful it is shown, using similar techniques to the ones by Kézdy and Snevily [5], that T decomposes K2hn;2hn for some h dr=4e, where r is the radius of T . Moreover, if the base tree of T is bigraceful or if there is a vertex v in T such that jViðvÞj

On Isoperimetric Connectivity in Vertex-Transitive Graphs

We shall define the

Covering a Finite Abelian Group by Subset Sums

k

On Restricted Sums

-isoperimetric connectivity

Vertex-transitive graphs that remain connected after failure of a vertex and its neighbors

\lambda _k

An Isoperimetric Problem in Cayley Graphs

of a regular graph