The aim of this paper is to characterize 3-dimensional almost Kenmotsu manifolds with ξ belonging to the (k, μ)′-nullity distribution and h′ 6= 0 satisfying certain geometric conditions. Finally, we give an example to verify some results.

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...

A banded invertible matrix has a remarkable inverse. All “upper” and “lower” submatrices of have low rank (depending on the bandwidth in ). The exact rank condition is known, and it allows fast multiplication by full matrices that arise in the boundary element method. We look for the “right” proof of this property of . Ultimately it reduces to a fact that deserves to be better known: Complement...

A graph G is singular of nullity (> 0), if its adjacency matrix A is singular, with the eigenvalue zero of multiplicity . A singular graph having a 0-eigenvector, x, with no zero entries, is called a core graph.We place particular emphasis on nut graphs, namely the core graphs of nullity one. Through symmetry considerations of the automorphism group of the graph, we study relations among the en...

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...

A simple digraph describes the off-diagonal zero-nonzero pattern of a family of (not necessarily symmetric) matrices. Minimum rank of a simple digraph is the minimum rank of this family of matrices; maximum nullity is defined analogously. The simple digraph zero forcing number is an upper bound for maximum nullity. Cut-vertex reduction formulas for minimum rank and zero forcing number for simpl...

We establish the bounds 43 6 bν 6 bξ 6 √ 2, where bν and bξ are the NordhausGaddum sum upper bound multipliers, i.e., ν(G)+ν(G) 6 bν |G| and ξ(G)+ξ(G) 6 bξ|G| for all graphs G, and ν and ξ are Colin de Verdière type graph parameters. The Nordhaus-Gaddum sum lower bound for ν and ξ is conjectured to be |G| − 2, and if these parameters are replaced by the maximum nullity M(G), this bound is calle...

The zero forcing number Z(G) is used to study the minimum rank/maximum nullity of the family of symmetric matrices described by a simple, undirected graph G. The positive semidefinite zero forcing number is a variant of the (standard) zero forcing number, which uses the same definition except with a different color-change rule. The positive semidefinite maximum nullity and zero forcing number f...