Depth-First Search Algorithms for Algebraic Connectivity Maximizing Regular Graphs

The algebraic connectivity is an indicator of how well connected a graph is. It also characterizes the convergence speed of some dynamic processes over networks. In this paper, taking into account that homogeneous networks are modeled as regular graphs, we consider the following problem: given a pair (n, k) of positive integers such that k is less than n and kn is an even number, find a k-regular graph with n vertices that have the maximum algebraic connectivity. We present depth-first search algorithms for solving this problem, which are exhaustive search but prune unnecessary search branches based on some known properties of the regular graph and the algebraic connectivity. We also show the results of execution of the proposed algorithms for the values of n up to 11.