Abstract—The nullity η(G) of a graph is the occurrence of zero as an eigenvalue in its spectra. A zero-sum weighting of a graph G is real valued function, say f from vertices of G to the set of real numbers, provided that for each vertex of G the summation of the weights f(w) over all neighborhood w of v is zero for each v in G.A high zero-sum weighting of G is one that uses maximum number of non-zero independent variables. If G is graph with an end vertex, and if H is an induced subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then, η(G)= η(H). In this paper, a high zero-sum weighting technique and the endvertex procedure are applied to evaluate the nullity of t-tupple and generalized t-tupple graphs are derived and determined for some special types of graphs, Also, we introduce and prove some important results about the t- tupple coalescence, Cartesian and Kronecker products of nut graphs. Keywords—Graph theory, Graph spectra, Nullity of graphs. I. INTRODUCTION HE eigenvalues of the adjacency matrix A(G) are said to be the eigenvalues of the graph G, the occurrence of zero as an eigenvalue in the spectrum of the graph G is called the “nullity” of G, it is denoted by η(G). Brown and others [2] proved that a graph G is singular if, and only if, G possesses a non-trivial zero-sum weighting, and asked, what causes a graph to be singular and what the effects of this on its properties are. Rashid [6] proved that the maximum number of non zero independent variables used in a high zero- sum weighting for a graph G, is equal to the nullity of G. Definition 1:[2],[6, p.16] A vertex weighting of a graph G is a function f: V(G)→R where R is the set of real numbers, which assigns a real number (weight) to each vertex. The weighting of G is said to be non-trivial if there is at least one vertex v∈V(G) for which f(v) ≠ 0. Definition 2:[2, p.16] A non-trivial vertex weighting of a graph G is called a zero-sum weighting provided that for each v∈V(G), ∑f(w) = 0, where the summation is taken over all w∈N G (v). Clearly, the following weighting for G is a non-trivial zero- sum weighting where x 1 , x 2 , x 3 , x 4 , and x 5 are weights and provided that (x 1 , x 2 , x 3 , x 4 , x 5 ) ≠ (0, 0, 0, 0, 0)as indicated in Fig.1. Kh.R. Sharaf is in University of Zakho-Department of Mathematics, Kurdistan Region of Iraq (e-mail: Khidirsharaf@yahoo.com). D.A.Ali is in University of Zakho-Department of Mathematics, Kurdistan Region of Iraq (e-mail: didarmath@yahoo.com). Fig. 1 A non-trivial zero-sum weighting for a graph G Theorem 1: A graph G is Singular if, and only if there is a non-trivial zero sum weighting for G.■ Out of all zero-sum weightings of a graph G, a high zero-sum weighting of G is one that uses maximum number of non-zero independent variables. Lemma 1: [6, p.35] In any graph G, the maximum number MV(G) of non zero independent variables in a high zero-sum weighting equals the number of zeros as an eigenvalues of the adjacency matrix of G, (i.e.MV(G)= η(G)).■ In Fig.1, the weighting for the graph G is a high zero-sum weighting that uses 5 independent variables, hence, η(G) = 5. This is a very active method to characterize the degree of singularity (nullity) of a chemical compound Graph, the carbohydrate graph C n H 2n+1 , with n=5, has two bonding graphs, (a) where the 5 carbon atoms induces a path of order 5, η(G) = 7 this is a more stable case which is usually present in the nature, while in (b) where the 5 carbon atoms induces a star of order 5, with η(G) = 9 which has different physical properties as in case a,as well as more instability. Lemma 2:[1, p.72],[3] i. The eigenvalues of the cycle C p are of the form 2cos p r 2π , r = 0, 1,…,p-1. According to this, η(C P )= 2 if p=0(mod4) and 0 otherwise. ii. The eigenvalues of the path P p are of the form 2cos 1 p r + π , r =1,2, … p. And thus, η(P P )=1 if p is odd and 0 otherwise. iii. The spectrum of the complete graph K p , consists of p-1 and -1 with multiplicity p-1. iv. The spectrum of the complete bipartite graph K m,n , consists of √ , - √ and zero m+n-2 times Lemma 3: (Endvertex Lemma)[4, p.234] If G is graph with an end vertex, and if H is an induced subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then η(G) = η(H). ■ Khidir R. Sharaf, Didar A. Ali Nullity of t-Tupple Graphs T World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:8, No:2, 2014 314 International Scholarly and Scientific Research & Innovation 8(2) 2014 scholar.waset.org/1307-6892/9997480 International Science Index, Mathematical and Computational Sciences Vol:8, No:2, 2014 waset.org/Publication/9997480