The Laplacian Spectrum of the Generalized $n$-Prism Networks

Abstract

‎The Laplacian eigenvalues and polynomials of the networks play an essential role in understanding the relations between the topology and the dynamic of networks‎. ‎Generally‎, ‎computation of the Laplacian spectrum of a network is a hard problem and there are just a few classes of graphs with the property that their spectra have been completely computed‎. ‎Laplacian spectrum for $ n$-prism networks was investigated in [Liu et al.‎, ‎Neurocomputing 198 (2016) 69-73]‎. ‎In this paper‎, ‎we give a method for calculating the eigenvalues and characteristic polynomial of the Laplacian matrix of a generalized $n$-prism network‎. ‎We show how such large networks can be constructed from small graphs by using graph products‎. ‎Moreover‎, ‎our results are used to obtain the Kirchhoff index and the number of the spanning trees in the generalized $n$-prism networks‎. ‎We also give some examples of applications‎, ‎that explain the usefulness and efficiency of the proposed method‎.