Computing the Kirchhoff index of a family of phenylene chain networks

Abstract The Kirchhoff index is a fundamental topological metric that provides insights into the structural and electrical characteristics of networks. It is defined as the sum of resistance distances between all pairs of nodes, serving as a key factor in understanding the dynamics within networks. To investigate the impact of structural variations on the Kirchhoff index, we select a family of phenylene chain networks as our model and establish a methodology to explore the Kirchhoff index using the Laplacian spectrum. By analyzing the network structure, we introduce a parameter to control the number of iterations, providing a recursive relationship between the Laplacian matrix and its eigenvalues at intervals of generations. This approach enables the derivation of an analytical expression for both the sum of the reciprocals of all nonzero Laplacian eigenvalues and the Kirchhoff index.