Narrow Results

In this paper, we investigate the random walks on metro systems in 28 cities from worldwide via the Laplacian spectrum to realize the trapping process on real systems. The average trapping time is a primary description to response the trapping process. Firstly, we calculate the mean trapping time to each target station and to each entire system, respectively. Moreover, we also compare the average trapping time with the strength (the weighted degree) and average shortest path length for each station, separately. It is noted that the average trapping time has a close inverse relation with the station’s strength but rough positive correlation with the average shortest path length. And we also catch the information that the mean trapping time to each metro system approximately positively correlates with the system’s size. Finally, the trapping process on weighted and unweighted metro systems is compared to each other for better understanding the influence of weights on trapping process on metro networks. Numerical results show that the weights have no significant impact on the trapping performance on metro networks.

The generalized windmill graphs are good models for many real-world networks. In this paper, we obtain analytic expressions for the eigenvalues of the adjacency matrices and of the Laplacian matrices of the generalized windmill graphs. Using this information, we study some structural and dynamical properties of these graphs. To the end, we investigate the metro networks of four France cities and propose our suggestions for the planning of public transport networks.