Panconnectivity Algorithm for Eisenstein-Jacobi Networks

Eisenstein-Jacobi (EJ) networks were proposed as an efficient interconnection network topology for parallel and distributed systems. They are based on Eisenstein-Jacobi integers modulo $a = a+b\rho$, where $0 \leq a \leq b$, and they are 6-regular symmetric networks and considered as a generalization of hexagonal networks. Most of the interconnection networks are modeled as graphs where the applications and functions of graph theory could be applied to. The cycles in a graph are one type of communications in interconnection networks that are considered as a factor to measure the efficiency and reliability of the networks' topology. The network is said to be panconnected if there are cycles of length $l$ for all $l = D(u, v), D(u, v)+1, D(u, v)+2, \dots, n-1$ where $D(u, v)$ is the shortest distance between nodes $u$ and $v$ in a given network. In this paper, we investigate the panconnectivity problem in Eisenstein-Jacobi networks. The proposed algorithm constructs and proves the panconnectivity of a given Eisenstein-Jacobi network and its complexity is $O(n^4)$. Simulation results are given to support the correctness of this work.