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 ...

We propose the theory of Cayley graphs as a framework to analyse gate counts and quantum costs resulting from reversible circuit synthesis. Several methods have been proposed in the reversible logic synthesis literature by considering different libraries whose gates are associated to the generating sets of certain Cayley graphs. In a Cayley graph, the distance between two vertices corresponds t...

The graph Laplacian plays key roles in information processing of relational data, and has analogies with the Laplacian in differential geometry. In this paper, we generalize the analogy between graph Laplacian and differential geometry to the hypergraph setting, and propose a novel hypergraph pLaplacian. Unlike the existing two-node graph Laplacians, this generalization makes it possible to ana...

With the advent of big data and information age, magnitude various complex networks is growing rapidly. Many real-life situations cannot be portrayed by ordinary networks, while hypergraphs have ability to describe characterize higher order relationships, which attracted extensive attention from academia industry in recent years. Firstly, this paper described development process, application ar...

It is, however, not clear how to choose the generators to produce special graphs. We know many topologies and their generators, but many more may be constructed in the future, having better properties (in terms of diameter, nodal degree and connectivity) than for instance the hypercube. I will present several graphs which connect rings using the generator g 1 and some additional generators.

The expansion G of a graph G is the 3-uniform hypergraph obtained from G by enlarging each edge of G with a new vertex disjoint from V (G) such that distinct edges are enlarged by distinct vertices. Let ex3(n, F ) denote the maximum number of edges in a 3-uniform hypergraph with n vertices not containing any copy of a 3-uniform hypergraph F . The study of ex3(n,G ) includes some well-researched...

For unbounded operators t, s between Hilbert C*-modules which admit the polar decompositions V|t|, W|s|, respectively, we obtain an explicit upper bound estimate for the gap between t and s in terms of the norm of the bounded operators V − W , C|t| − C|s| and C|t∗| − C|s∗|, where C|t| and C|s| are the Cayley transforms of |t| and |s|. The result are used to drive a criterion for continuity of t...

Given a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is ...