The purpose of this paper is to show how eigenvalue is used in the field of graph theory. Earlier, the theory of eigenvalues and their maximum eigenvalue of the adjacency matrix emerging from a graph has been ascertained. Here we examine the relationship between the adjacency matrix, eigen values and spectrum of 3- regular graphs with vertices 4, 5, 6, 7 and 8. Also we have examined the relationship between the adjacency matrix, eigen values and spectrum of 4-regular graphs with vertices 5, 6, 7 and 8.