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.

Coalescing Fiedler and core vertices

“…In [18], [19], Sciriha determined properties of substructures responsible for a graph to be singular, establishing that in a graph of nullity η, there are η induced subgraphs (termed singular configurations) of nullity one from a prescribed list. In [22], Sharaf and Ali proceeded to determine a sharp lower bound and a sharp upper bound for the nullity of the coalescence of two graphs. They showed that the nullity of the coalescence of s component graphs varies by at most s − 1 from the sum of the nullities of the component graphs.…”

Change of nullity of a graph under two operations

“…A non-trivial vertex weighting of a graph G is called a zero sum weighting provided that for each v∈V (G), u∈N(v)f(u) = 0, that is the summation is taken over all u∈N(v), See (Brown et al, 1993). Out of all zero-sum weightings of a graph G, a high zero sum weighting (hzsw) of G, is one that uses a maximum number of non-zero independent variables, see (Sharaf & Ali, 2013). The nullity, η(G of a graph G) is the multiplicity of zero as an eigenvalue of its adjacency matrix, see (Cheng & Liu, 2007)., So a graph is singular if its nullity is at least one.…”

“…The nullity of most known graphs can be easily determined using the hzsw tool, since number of independent variables used in a hzsw of the graph is exactly η(G). Thus, η(Kn)=0, for n≥2, η(Km,n)=m +n-2, η(Pn)=1 if n is odd and zero where n is even, η(Cn)=2 if n=0 mod4 and it is non-singular otherwise, see (Cheng, & Liu, 2007) and (Sharaf & Ali, 2013). Maximum nullity of graphs was studied in (Gutman & Sciriha, 1996).…”