Undirected Graphs Via Spectrum and Some of Applications رسالة ماجستير

Abstract

The algorithms of listing triangles of a graph @ have wide applications in many areas and their solutions are of great importance in solving many problems. For example, the complex graph analysis has measures that directly compute using the triangle listing results such as the computation of cluster coefficient. For a simple undirected graph G, there are algorithms to list triangles that iterate over the vertices or usc all the entries of the associated adjacency matrix A. In this work, we present two new algorithms: Algorithm l and Algorithm 2, to list all triangles of a graph G that depend on the nonzero upper triangle entries of the adjacency matrix and their corresponding indices. In addition, we present two new theorems for calculating the number of triangles for a complete graph Jn. In the chemical application, there is a relation between the Laplacian and the adjacency spectrum with the maximum degree of the isomers of an alkane CnHan» 4 € n 12. To illustrate the relation, we use the software to find the values of the spectrum and the lopological index W of the isomers of the

n-alkane, n = 4, 5, 6, 7, 8.