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On the Nullity of Expanded Graphs Gen. Math. Notes, Vol. 21, No. 1, March 2014, pp. 97-117 ISSN 2219-7184; Copyright © ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in On the Nullity of Expanded Graphs Khidir R. Sharaf 1 and Kavi B. Rasul 2 1,2Department of Mathematics, Faculty of Science University of Zakho, Zakho, Iraq 1E-mail: [email protected]; [email protected] 2E-mail: [email protected] (Received: 13-12-13 / Accepted: 22-1-14) Abstract The nullity (degree of singularity) η(G) of a graph G is the multiplicity of zero as an eigenvalue in its spectrum. It is proved that, the nullity of a graph is the number of non-zero independent variables in any of its high zero-sum weightings. Let u and v be nonadjacent coneighbor vertices of a connected graph G, then η(G) = η(G−u) + 1 = η(G−v) + 1. If G is a graph with a pendant vertex (a vertex with degree one), and if H is the subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then η(G) = η(H). Let H be a graph of order n and G 1, G 2,…, G n be given vertex disjoint graphs, then the expanded G i graph H n is a graph obtained from the graph H by replacing each vertex v i of H by a graph G i with extra sets of edges S i,j for each edge v ivj of H in which S i,j = {uw: u ∈V(G i), w ∈V(G j)}. In this research, we evaluate the nullity of expanded graphs, for some special ones, such as null graphs, complete bipartite graphs, star graphs, complete graphs, nut graphs, paths, and cycles. Keywords: Graph Theory, Graph Spectra, Nullity of a Graph. I Introduction A graph G is said to be a singular graph provided that its adjacency matrix A(G) is a singular matrix. The eigenvalues λ1, λ2, … λp of A(G) are said to be the 98 Khidir R. Sharaf et al. eigenvalues of the graph G, which form the spectrum of G. The occurrence of zero as an eigenvalue in the spectrum of the graph G is called its nullity (degree of singularity) and is denoted by η(G). See [1] and [3]. Definition 1.1[2, 5, 7] A graph G is said to be η -singular or the nullity of G is η , abbreviated η(G) or η if, the multiplicity of zero (as an eigenvalue) in S p(G) is η . Definition 1.2[2] 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. A 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 1.3[2] A non-trivial vertex weighting of a graph G is called a zero-sum weighting provided that for each v ∈V(G), ∑f( u ) = 0, where the summation is taken over all u ∈NG(v). Clearly, the following weighting for G is a non-trivial zero-sum weighting, where x and y are weights and (x, y) ≠ (0, 0), as indicated in Fig.1.1. x x . y -y y - x -x Figure 1.1: A non-trivial zero-sum weighting of a graph Definition 1.4[5] 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, M v(G) . An important relation between the singularity of a graph, and existence of a zero- sum weighting is, that a graph is singular iff it possesses a non trivial zero-som weighting.[2] Proposition 1.5[5] In any graph G, the maximum number 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. In Fig. 1.1, the weighting for the graph G is a high zero-sum weighting that uses 2 independent variables, hence, η(G) = 2. Let r(A(G)) be the rank of A(G). Clearly, η(G) = p – r(A(G)). The rank r(G) of a graph G is the rank of its adjacency matrix A(G). Then, each of η(G) and r(G) determines the other. On the Nullity of Expanded Graphs 99 The nullity of some known graphs such as cycle Cn , path P n, complete K p and complete bipartite K r,s graphs are given in the next lemma. Lemma 1.6[5, 6, 7] i) The eigenvalues of the cycle C n are of the form: 2π r 2cos , r = 0, 1, …, n-1. According to this, n 2, if n = 0(mod 4), η(Cn ) = 0, otherwise. ii) The eigenvalues of the path P n are of the form: π r 2cos , r =1,2, … n. And thus, n +1 1, if n is odd, η(Pn ) = 0, if n is even. iii) The nullity of the complete graph K p, is: ,1 if p = ,1 η = (K p ) ,0 if p > .1 iv) The nullity of the complete bipartite graph K r,s , is: 0 if r = s = 1, η(Kr,s ) = r + s - 2 ot herwi se . Definition 1.7: Two vertices of a graph G are said to be of the same type (coneighbors) if they are not adjacent and have the same set of neighbors. Thus, the two vertices v i, v j of the same type have the same row vectors R i = R j th th describing them, where R i and R j are the i and j row vector of A(G), corresponding to the vertices vi and v j , i, j = 1, 2, …, p. Each pair of such (same type) vertices results in two dependent (coincide) rows which yield a zero in spectra of the graph G. It is clear that the occurrence of m equal rows contributes (m-1) to the nullity. Corollary 1.8[4] (End Vertex Corollary): If G is a graph with a pendant vertex, and H is the subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then η(G) = η(H). 100 Khidir R. Sharaf et al. Applying Corollary 1.8, several times, deleting v 1 and v 3 respectively is illustrated in the next figure. v1 v2 v3 η( ) v6 v4 v5 v7 v3 = η( ● ) v4 v v 6 v7 5 = η(● ● ●) =3 Figure 1.2: Illustration of Corollary 1.8 So (End Vertex Corollary) is a strong tool to determine the nullity of trees. Operations on Graphs Many interesting graphs are obtained from combining pairs (or more) of graphs or operating on a single graph in some way. We now discuss a number of operations which are used to combine graphs to produce new ones. Lemma 1.9[1, 3] Let G = G 1∪G2∪…∪Gt, where G 1, G 2, …, G t are connected t η= η components of G, then ()G∑ () G i i =1 Definition 1.10[1, 3] The join G1+G 2 of two graphs G 1 and G 2 is a graph whose vertex set, V(G 1+G 2) = V(G 1)∪V(G 2), and edge set E(G) = E(G 1)∪E(G 2) ∪{uv: for all u ∈G1 and all v ∈G2}. Definition 1.11[1] The sequential join G1+G 2+ … +G n of n disjoint graphs G1,G 2,…,G n is the graph (G 1+G 2)∪(G 2+G 3)∪…∪(G n-1+G n) and denoted by n ∑Gi , for i=1,2,…,n and i =1 defined by n n n n ∪ V( ∑Gi ) = U V ( G i ) , E( ∑Gi )={ U E( G i ) } i =1 i =1 i =1 i =1 {uv: for all u ∈Gi and all v ∈ Gi+1 , i=1,2,…,n-1}. On the Nullity of Expanded Graphs 101 As depicted in Fig.1.3. … G1 G2 Gn-1 Gn n Figure 1.3: The sequential join graph ∑Gi i =1 n n n n n −1 + It is clear that, p( ∑Gi ) = ∑ pi , q( ∑Gi ) = ∑qi ∑ p i p i +1 , i =1 i =1 i =1 i=1 i = 1 In which p i = p(G i) and q i = q(G i). Definition 1.12[4] Let P n be a path with vertex {v 1, v 2,…,v p}. Replacing each vertex v i by an empty graph N of order p i, for i=1,2,…,p and joining edges pi between each vertex of N and each vertex of N for i=1,2,…,n–1, we get a pi pi +1 n graph order p 1+p 2+…+p n, denoted by ∑ N . Such graph is called a sequential pi i =1 join. Definition 1.13[1] The strong product graph G1⊠G2 of G 1 and G 2, is the union of the Cartesian product graph G1×G2 and the Kronecker product graph G1⨂G2. Clearly, p(G 1⊠G2) = p(G 1)p(G 2) and q(G 1×G 2) = p(G 1)q(G 2)+ p(G 2) q(G 1) + 2 q(G 1) q(G 2). It is apparent that K m ⊠ Kn =K mn . Results, relating the nullity of the graphs G1 and G2 and their strong product G1⊠G2, are not studied widely.
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