Construction and Nullity of Some Classes of Smith Graphs

For the adjacency matrix A of a graph G, a number λ is an eigenvalue of G if for some non zerovector X, AX=λX. The vector X is called the eigenvector corresponding to λ. The eigenvalues are exactly those numbers λ that make the matrix A-λI to be singular. All eigenvectors corresponding to λ forms a subspace Vλ; the dimension of Vλ is equal to the multiplicity of λ. A graph G is a Smith graph if 2 is an eigenvalue of the adjacency matrix A of G, a λ-weighting technique is introduced and applied to characterize some classes of Smith graphs as well as to study their nullities and the nullity of vertex identification of such graphs. We also have proved that under certain conditions the vertex identification of some Smith graphs is a Smith graph.