Spectral characterization of some graphs

AbstractThe lexicographic product (or composition) of a cycle of length r with a totally disconnected graph on n vertices is shown to be a graph characterized by the eigenvalues of its adjacency matrix for all odd r

Cayley DHTs - A Group-Theoretic Framework for Analyzing DHTs Based on Cayley Graphs

Static DHT topologies influence important features of such DHTs such as scalability, communication load balancing, routing efficiency and fault tolerance. Nevertheless, it is commonly recognized that the primary difficulty in designing DHT is not in static DHT topologies, but in the dynamic DHT algorithm which adapts various static DHT topologies to a dynamic network at Internet. As a direct consequence, the DHT community has been paying more attention to the dynamic DHT algorithm design, resulting in a variety of DHT systems lacking of a common view for analysis and interoperation.In this paper we reiterate the importance of static DHT topologies in the DHT system design by analyzing and classifying current DHTs in terms of their static topologies based on a grouptheoretic model: Cayley graphs. We show that most of current DHT proposals use Cayley graphs as static DHT topologies, thus taking advantage of several important Cayley graph properties such as vertex/edge symmetry, decomposability, optimal fault tolerance and hamiltonicity. We observe that several non-Cayley-graph based DHT proposals such as Koorde /D2B/Distance Halving and Pastry/Tapestry also rely on techniques in their dynamic DHT algorithm design trying to imitate desirable Cayley graph properties. Based on Cayley graphs, we propose the class of Cayley DHTs as a unified group-theoretic model for investigating DHTs from a graph theoretic perspective. The significance of Cayley DHTs is in their explicit inspiration to a uniform dynamic DHT algorithm design, which can directly leverage algebraic design methods thus is able to generate sets of highperformance DHTs adopting various Cayley graph based static DHT topologies but still sharing the same dynamic DHT algorithm