Linearisation of graphs using walks of unit length

Abstract

An algorithm using walks of unit length has been developed for transforming arbitrarily weighted graphs into isospectral linear chains (with weighted vertices and edges). Eigenspectral features of some regular graphs such as complete graphs, cycles and stars have been shown to be generated by a single-step operation of the algorithm. Recursive relationships for the coefficients of characteristic polynomials (CPs) of weighted linear chains have been given, and it has been shown that CPs of complicated chemical graphs and graphs with arbitrary edge, and vertex, weights can be constructed fairly easily using such relationships, after they have been linearised by the algorithm.