A review on eigen values of adjacency matrix of graph with cliques

Abstract

The paper reviews the applications of eigen value in different areas. One of the area is in the analysis of graphs coming from networks. The development of theory regarding the eigenvalues and its maximum eigenvalue of the adjacency matrix arising from a general graph is already well-established. Here we review some notions from different context, i.e. led by observation of some simple experiments regarding the relation between graph, cliques, and the eigen values of the adjacency matrix. We focus on regular graphs having one or more cliques in their graph structures. We do some numerical experiment on the computation of the eigen values of the adjacency matrix and show some patterns on the relation between the structure of the graph (e.g. the maximum cliques, chromatic number) and the eigen values of the adjacency matrix. By observing these patterns we find some conclusion, such as: i. the maximum clique of a complete graph is given by the largest eigen value plus one; ii. the maximum clique of a cycle g...