The clustering cofficient C of a network, Which is a measure of direct conuectivity between neighbors of the various nodes, ranges from O (for no connectivity) to 1 (for full connectivity). We define ex-tended clustering coefficients C (h) of a small-world network based on nodes that are at distance h from a source node, thus generalizing distance-1 neighborhoods employyed in computing the ordinary cluster-ing coefficient C = C(1). Based on known results about the distance distribution Pb(h) in a network, that is, the probability that a randomly chosen pair of vertices have distance h, we derive and experimentally validate the law Ps(h)C(h)< clogN/N, where c is a small constant that seldom exceeds 1. This result is significant because it shows that the product Ps(h)C(h) is upper-bouded by a value that is considerably smaller than the product of maximum values for Pb(h) and C(h).

In this note we study the linear symmetry group LS(f) of a Boolean function f of n variables, that is, the set of all £ GLn(2) which leave f invariant, where GLn (2) is the general linear group on the field of two elements. The main problem is that of concrete reproesentation: which subgroups G of GLn(2) can be represented as G=LS(f) for some n-ary k=valued Boolean function f. We call such subgroups linearly representable. The main results of the note may be summarized sa follows: We give a necessary and sufficient condition that a subgroup of GLn(2) is linearly representable and obtain some results on linear representability of its subgroups. Our results generalize some theorems from P.Clote and E. Kranakis [SIAMJ. Comut. 20 (1991) 553-590]; A Kisielewicz [J. Algebra 199 (1998) 379-4031].

Despite numerous interconnection schemes proposed for distributed multicomputing, systematic studies of classes of inter-processor networks, that offer speed-cost tradeoffs over a wide range, have been few and far in between. A notable exception is the study of Cayley graphs that model a wide array of symmetric networks of theoretical and practical interest. Properties established for all, or for certain subclasses of, Cayley graphs are extremely useful in view of their wide applicability. In this paper, we obtain a number of new relationships between Cayley (di) graphs and their subgraphs and coset graphs with respect to subgroups, focusing in particular on homomorphism between them and on relating their internode distances and diameters. We discuss applications of these results to well-known and useful interconnection networks such as hexgonal and honeycomb meshes as well as certain classes of pruned tori.

In this paper, a new routing algorithm is given for the shuffle-exchange permutation network (SEPn). The length of the path between any two nodes given by our algorithm is not more than 11/16n2+O(n), Le., the diameter of SEPn is most 11/16n2+O(n). This improves on a 1/8 (9n2-22n+24) routing algorithm described earlier by S. Latifi and P. K. Srimani. We also show that the diameter of SEPn is more than 2n2-n.

In this paper, we extend known relationships between Cayley digraphs and their subgraphs and coset graphs with respect to subgroups to obtain a number of general results on homomorphism between them. Intuitively, our results correspond to synthesizing altemative, more economical, interconnection networks by reducing the number of dimensions and/or link density of existing networks via mapping and pruning. We discuss applications of these results to well-known and useful interconnection netorks such as hexagonal and honeycomb meshes, including the derivation of provably correct shortest-path routing algorithems for such networks.