On some properties of a fixed-degree cayley graph from linear group

Abstract

Cayley graph is a mathematical model in which all vertices and edges are represented as elements of a finite group $G$ and relations between the elements with respect to some generating set $S$ of $G$. Since the model is based on rich background of group theory, it has many advantages: the symmetric structures in any Cayley graph make itself preferable as a static interconnection network. Furthermore, sensible choice of its basis group and corresponding generating set can add more desirable properties. So it is widely used in designing and analyzing interconnection networks. This thesis proposes a new class of Cayley graph based on the linear group $\mbox{PSL}(2,p)$ for a prime power integer $p$, and considers some of its properties. The graph has $O(p^3)$ nodes and fixed degree 3. Similar to the structure of cube-connected cycle, it consists of interconnected cycles. We show some additional properties of the graph. It is planar for $p=3,5$ and has maximum connectivity. The hamiltonicity of the graph is also considered. Finally, a simple routing algorithm on the graph is developed, which bounds the diameter of the graph to $O(\sqrt[3]{|V|})$. By experimental results, we conjecture that the diameter of the graph is $O(\log |V|)$ in optimum.