Spectral analysis for weighted iterated <i>q</i>-triangulation networks

Abstract

Deterministic weighted networks have been widely used to model real-world complex systems. In this paper, we study the weighted iterated q-triangulation networks, which are generated by iteration operation F(⋅). We add q(q∈N₊) new nodes on each old edge and connect them with two endpoints of the old edge. At the same time, the newly linked edges are given weight factor r(0<r≤1). From the construction of the network, we obtain all the eigenvalues and their multiplicities of its normalized Laplacian matrix from the two successive generations of the weighted iterated q-triangulation network. Further, as applications of spectra of the normalized Laplacian matrix, we study the Kemeny constant, the multiplicative degree-Kirchhoff index, and the number of weighted spanning trees and derive their exact closed-form expressions for weighted iterated q-triangulation networks.