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].

Hypercube and Kautz network each possess certain desirable properties. However, some of the attractive features of one network are not found in the other. A novel class of network topologies proposed in this paper has the generalized hypercube and the Kautz network as its two extremes. The propsed network inberits the topological properties of both the Kautz network and extremes. The Propsed network inherits the topological properties of both the Kautz network and the generalized hypercube to a varying degree. This allows us to trade-off cost and performance effectively and construct networks which are most suitable for a particular purpose. In the present paper, we investigate the connectivity, wide-diameter, fault-tolerance, Hamiltonicity.

Hexagonal mesh and torus, as well as honeycomb and certain other pruned torus networks, are known to belong to the class of Cayley groups which are node-symmetric and possess other interesting mathematical properties. In this paper, we use Cayley-graph formulations for the aforementioned networks, along with some of our previous results on subgraphs and coset graphs, to draw conclusions relating to internode distance and network diameter. We also use our results to refine, clarify, and unify a number of previously published properties for these networks and other networks derived from them.

得到有限可解群及其某些子群的阶与其Fitting 子群的阶之间的若干关系。

We consider the relationships between Cayley digraphs and their coset graphs with respect to subgroups and obtain some general results on homomorphism and broadcasting between them. We also derive a general factorization theorem on subgraphs of Cayley digraphs by their automorphism groups. We discuss the applications of these results to well-known interconnection networks such as the butterfly network, the de Bruijn network, the cube-connected cycles network and the shuffle-exchange network.