Nullity and Bounds to the Nullity of Dendrimer Graphs

ABSTRACT In this paper, a high zero-sum weighting is applied to evaluate the nullity of a dendrimer graph for some special graphs such as cycles, paths, complete graphs, complete bipartite graphs and star graphs. Finally, we introduce and prove a sharp lower and a sharp upper bound for the nullity of the coalescence graph of two graphs.

iii) The spectrum of the complete graph Kp, consists of p-1 and -1 with multiplicity p- 1. iv) The spectrum of the complete bipartite graph Km,n , consists of mn, -mn and zero m+n-2 times Corollary 2.8: [4, p.234] If G is a bipartite 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). ■ Corollary 2.9: [4, p.235] Let G1 and G2 be two bipartite graphs in which η(G1) = 0. If the graph G is obtained by joining an arbitrary vertex of G1 by an edge to an arbitrary vertex of G2, then η(G) = η(G2).■

Coalescence Graphs
To identify nonadjacent vertices u and v of a graph G is to replace the two vertices by a single vertex incident to all the edges which are incident in G to either u or v. Denote the resulting graph by G{u, v}. To contract an edge e of a graph G is to delete the edge and then (if the edge is a link) identify its ends. The resulting graph is denoted by Ge. Definition 2.10: [7] Let (G1, u) and (G2, v) be two graphs rooted at vertices u and v, respectively. We attach G1 to G2 (or G2 to G1) by identifying the vertex u of G1 with the vertex v of G2. Vertices u and v are called vertices of attachment. The vertex formed by their identification is called the coalescence vertex. The resulting graph G1  G2 is called the coalescence (vertex identification) of G1 and G2. Definition 2.11: [7] Let {(G1, v1), (G2, v2), …, (Gt, vt)} be a family of not necessary distinct connected graphs with roots v1, v2, …, vt, respectively. A connected graph G= G1  G2  …  Gt is called the multiple coalescence of G1, G2,…,Gt provided that the vertices v1, v2, …, vt are identified to reform the coalescence vertex v. The t-tupple coalescence graph is denoted by t G is the multiple coalescence of t isomorphic copies of a graph G. In the same ways 2 1 t GG is the multiple coalescence of G1 and t copies of G2. Remark 2.12: [7] All coalescened graphs have v as a common cut vertex. Some graphs and their operations will, herein, be illustrated in Figure 2.2. Definition 2.13: [7] Let G be a graph consisting of n vertices and L = {H1, H2, …, Hn} be a family of rooted graphs. Then, the graph formed by attaching Hk to the k-th (1  k  n) vertex of G is called the generalized rooted product and is denoted by G(L); G itself is called the core of G(L). If each member of L is isomorphic to the rooted graph H, then the graph G(L) is denoted by G(H). Recall G1, G2 and G3 from Figure 2.2. Then, we have Definition 2.14: [7] The generalization of the rooted product graphs is called the Fgraphs, which are consecutively iterated rooted products defined as: In general,

Nullity of Dendrimer Graphs
In this section, we determine the nullity of dendrimer graphs k Also, from the condition that , for all v in the central cycle n C 4 , we have, And, for 2 4 , ... , n n n C C C = , so each identification of a copy of 4n C with a vertex of 4n C adds (increases) one to the nullity of a dendrimer graph. Since 4n C has n 4 vertices; thus, n 4 copies of a cycle n C 4 are identified to 4n C . Therefore, 4 4 4 ( ( )) ( ) 4 2 4 . C is a dendrimer graph having n 4 cycles and each cycle has 1 4 − n vertices to be identified with new vertices, hence we attach a copy of 4n C to ) 1 4 ( 4 − n n vertices. Also, each copy of 4n C adds (increases) one to the nullity of a dendrimer graph. Therefore, Similarly, we have, , there exists no non-trivial zero-sum weighting for 42 , there exists no non-trivial zero-sum weighting for 41 n C − , ... Then, from the condition that , for all v in ... , 2 , 1 , have: Hence, from Equations (3.5) and (3.6), we get: Also, from the condition that , for all v in ... , 2 , 1 , Hence, from Equations (3.7), (3.8) and (3.9), we get Therefore, each vertex of 2 This means that there exists a non-trivial zero-sum weighting for 2 , is similar to that for k=2. iv) The proof is similar to that of part (ii).■ While, the diameter of k D depends on the choice of the rooted vertex. Also, the maximum degree will be either 3 or 4 for 2  k , while the minimum degree is 1 .      Then, from the condition that  ii) The proof is similar to that of Proposition 3.6.■

Upper Bounds for the Nullity of Coalescence Graphs
In this section, we shall introduce and prove a lower and an upper bound for the nullity of the coalescence graph   Moreover, strictly holds at the left side if both rooted vertices have zero weights in their high zero-sum weightings, because there exists a zero-sum weighting which is the union of both high zero-sum weightings of G1 and G2.
Equality holds at the right side if both rooted vertices are cut vertices with zero weights in their high zero-sum weightings, and each component obtained with a deleting of a rooted cut vertex is singular, because there exists a high zero-sum weighting for 2 1 G G  that uses an extra independent variable further than the variables used in high zero-sum weightings of G1 and G2. See