In this paper we take a close look at the nullity theorem as formulated by Markham and Fiedler in 1986. The theorem is a valuable tool in the computations with structured rank matrices, it connects ranks of subblocks of an invertible matrix A with ranks of other subblocks in his inverse A. A little earlier, Barrett and Feinsilver, 1981, proved a theorem very close to the nullity theorem, but re...

The spectrum of a graph G is the set of eigenvalues of the 0–1 adjacency matrix of G. The nullity of a graph is the number of zeros in its spectrum. It is shown that the nullity of the line graph of a tree is at most one. c © 2001 Elsevier Science B.V. All rights reserved.

The minimum rank of a sign pattern matrix is defined to be the smallest possible rank over all real matrices having the given sign pattern. The maximum nullity of a sign pattern is the largest possible nullity over the same set of matrices, and is equal to the number of columns minus the minimum rank of the sign pattern. Definitions of various graph parameters that have been used to bound maxim...

The number of self-dual cyclic codes of length pk over GR(p2,m) is determined by the nullity of a certain matrix M(pk, i1). With the aid of Genocchi numbers, we determine the nullity of M(pk, i1) and hence determine completely the number of such codes.

We show that the number of fundamentally different m-colorings of a knot K depends only on the m-nullity of K, and develop a formula for the number of such colorings. We also determine the m-colorability and m-nullity of any (p, q, r) pretzel knot, and therefore determine the number of fundamentally different m-colorings for any (p, q, r)

We study the stability of harmonic morphisms as a subclass of harmonic maps. As a general result we show that any harmonic morphism to a manifold of dimension at least three is stable with respect to some Riemannian metric on the target. Furthermore we link the index and nullity of the composition of harmonic morphisms with the index and nullity of the composed maps.

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...

Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.