THE OPERATION OF NULLITY DECREES

The Rank+Nullity Theorem

The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W , then dim(V ) = rank(T ) + nullity(T ), where rank(T ) = dim(im(T )) and nullity(T ) = dim(ker(T )). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T ) and extend it to a basis B of V , and then show that dim(im...

On the nullity of graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is not isomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n− 2 and the second upper bound n− 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is al...

Nullity of Measurement - induced

Decrees for Damages in New Jersey

It is not the purpose of this article to discuss all the situations in which damages are awarded by the New Jersey Court of Chancery, for all the decisions stem from one general rule, namely, that whenever a court of equity has once acquired jurisdiction of a cause, it will retain such cause in order to do full and complete justice between the parties with respect to the subject-matter. This ru...

Ela on the Nullity of Graphs

The nullity of a graph G, denoted by η(G), is the multiplicity of the eigenvalue zero in its spectrum. It is known that η(G) ≤ n − 2 if G is a simple graph on n vertices and G is not isomorphic to nK1. In this paper, we characterize the extremal graphs attaining the upper bound n− 2 and the second upper bound n− 3. The maximum nullity of simple graphs with n vertices and e edges, M(n, e), is al...