REMOVAL OF DEGENERACY IN THE TOPOLOGICAL INDICES OF COSPECTRAL GRAPHS USING EDGE WEIGHTED MOLECULAR GRAPHS

Abstract

Two graphs having the same number of vertices connected in the same way are said to be isomorphic. Topological matrices representing such graphs will have the same eigen value spectrum. Hence isomorphic graphs are also known as co spectral or iso spectral graphs.  There are certain molecular graphs representing chemically different substances which are found to be co spectral graphs. Therefore the molecular descriptors calculated using topological matrices will not be able to distinguish between these molecules.  Under such circumstances, edge weighted topological matrices can be formulated so that, their eigen value spectrum will be unique. The edge weighted molecular graphs with bond length as the weight for the edges in adjacency and distance matrices distinguish between the molecules having similar molecular graphs. So different molecules having the same molecular graph will have the same topological indices, but the edge weighted adjacency and distance matrices have the ability to overcome this problem of degeneracy.