Fiedler vector analysis for particular cases of connected graphs

In this paper, we consider a p-Laplacian eigenvalue boundary value problem involving both right Caputo and left Riemann-Liouville types fractional derivatives. To prove the existence of solutions, we apply the Schaefer’s fixed point theorem. Furthermore, we present the Lyapunov inequality for the corresponding problem.

Network Connectivity Control of Mobile Robots by Fast Position Estimations and Laplacian Kernel

Together with wireless distributed sensor technologies, the connectivity control of mobile robot networks has widely expanded in recent years. Network connectivity has been greatly improved by theoretical frameworks based on graph theory. Most network connectivity studies have focused on algebraic connectivity and the Fiedler vector, which constitutes a network structure matrix eigenpair. Theoretical graph frameworks have popularly been adopted in robot deployment studies; however, the eigenpairs’ computation requires quite a lot of iterative calculations and is extremely time-intensive. In the present study, we propose a robot deployment algorithm that only requires a finite iterative calculation. The proposed algorithm rapidly estimates the robot positions by solving reaction-diffusion equations on the graph, and gradient methods using a Laplacian kernel. The effectiveness of the algorithm is evaluated in computer simulations of mobile robot networks. Furthermore, we implement the algorithm in the actual hardware of a two-wheeled robot.