Kirchhoff index of Vicsek polygon networks and its applications

The Kirchhoff index is a novel distance-based topological index corresponding to networks, which is the sum of resistance distances between all pairs of nodes. It assumes a significant role in describing the flow of a network and can also characterize the stability of the network. The computation of the Kirchhoff index of a network is frequently performed through spectral analysis methods. However, for networks with irregular structures, this method may not be applicable. In this paper, we propose a polygon network model and calculate its Kirchhoff index by reconstructing the network construction process. Furthermore, by establishing the relationship between the known Kirchhoff index and the Laplacian spectrum of the network, we derive the Kirchhoff index of the network and its relationship with other network indices, such as the Global mean-first passage time and the average path length. We then perform calculations on these related indices to gain a more comprehensive understanding of the network.