Vol.09 Issue-02, (July - December, 2017) ISSN: 2394-9309 (E) / 0975-7139 (P) Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856) Double-Blind Peer Reviewed Refereed Open Access International e-Journal - Included in the International Serial Directories Aryabhatta Journal of Mathematics and Informatics http://www.ijmr.net.in email id- irjmss@gmail.com Page 141 On the Spectrum and Energy of Rooted Product of Graphs T. K. Mathew Varkey ฀ John K. Rajan † ฀ Department of Mathematics, TKM College of Engineering, Kollam 5, Kerala, India. † Department of Mathematics, University College, Thiruvananthapuram 34, Kerala, India. Abstract:A graph  with zero as an eigenvalue of its adjacency matrix is singular. Singular graph has three types of vertices: core vertices and noncore vertices of null spread 0 and -1. If  is a labelled graph on n vertices and G is a sequence of n rooted graphs  1 ,  2 ,...,   , the graph obtained by identifying the root of   with the i th vertex of H is the rooted product of  by . The sum of the absolute values of the eigenvalues of the adjacency matrix ()of the graph  is called the energy of the graph . Energy of a graph  is denoted by ().In this paperthe spectrum and energyof the rooted product of singular graphs having three types of vertices as roots were studied. An attempt was also made to study the spectrum of the corona product of two graphs using rooted product. Keywords: Core Vertices, Energy of graph, Nullity, Rooted Product, Spectrum. 1. Introduction Let G = (V(G),E(G)) be a finite, undirected simple graph with vertex set V(G) and edge set E(G). The number of vertices n of G is the order of G. A square matrix of order n whose entries a ij denote the number of edges from vertex v i to vertex vj is the adjacency matrix A(G) of the graph G. The characteristic polynomial of the adjacency matrix A(G), denoted by G(λ) is called the characteristic polynomial of the graph G . The roots of the equation G(λ ) = 0 are called the eigenvalues of the graph G. The collection of the eigenvalues together with their multiplicities constitute the spectrum of G and is denoted by spec(G). The graph G with zero as an eigenvalue is called a singular graph. The multiplicity of zero in the graph’s spectrum is the nullity η(G) of the graph G. A non-zero vector X satisfying the equation AX = 0 is called kernel eigenvector of G. Let G be a singular graph of nullity η and let X = [x 1 ,x 2 ,...,x m , 0, . . . ,0] T , where x i ≠ 0,i = 1,2,...,m be a kernel eigenvector. Thesub graphFofGinducedbythefirst mverticescorresponding to the non-zero entries x 1 ,x 2 ,...,x m is called the core of G. The remaining vertices are called core-forbidden vertices or noncore vertices. A singular graph on atleast two vertices, with a kernel eigenvector having non-zero entries is called a core graph(See Figure 1). Definition 1.1.[17]. If G is a graph and x V(G), then the neighbour set of x is defined by N(x) =    ∶   ( ). Theorem1.1.[17]. If a graph has two distinct vertices x and y such that N(x) = N(y) ,then G is singular.