A matroid is uniform if and only it has no minor isomorphic to U1,1?U0,1 paving U2,2?U0,1. This paper considers, more generally, when a M Uk,k?U0,?-minor for fixed pair of positive integers (k,?). Calling such (k,?)-uniform, shown that this equivalent the condition every rank-(r(M)?k) flat nullity less than ?. Generalising result Rajpal, we prove any (k,?) prime power q, finitely many simple co...

We investigate the complexity of counting trees, forests and bases of matroids from a parameterized point of view. It turns out that the problems of computing the number of trees and forests with k edges are #W[1]-hard when parameterized by k. Together with the recent algorithm for deterministic matrix truncation by Lokshtanov et al. (ICALP 2015), the hardness result for k-forests implies #W[1]...

the Kronecker sum of DD with itself, a total of d times. Using a standard fact about Kronecker sums, if ρ1, . . . , ρN denote the eigenvalues of DD then ρi1 + ρi2 + · · ·+ ρid , i1, . . . , id ∈ {1, . . . , N}, are the eigenvalues of (∆̃) ∆̃. By counting the multiplicity of the zero eigenvalue, we arrive at a nullity for ∆̃ of (k + 1). One can now directly check that each of the polynomials specif...

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [ABS00] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [ABS04b], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-null...

Generalizing a previous one-variable " interlace polynomial " , we consider a new interlace polynomial in two variables. The polynomial can be computed in two very different ways. The first is an expansion analogous to the state space expansion of the Tutte polynomial; the differences are that our expansion is over vertex rather than edge subsets, the rank of the subset appears positively rathe...

The aim of this work is to extend the results of S. Nayatani about the index and the nullity of the Gauss map of the Costa-Hoffman-Meeks surfaces for values of the genus bigger than 37. That allows us to state that these minimal surfaces are non degenerate for all the values of the genus in the sense of the definition of J. Pérez and A. Ros. Introduction In the years 80’s and 90’s the study of ...

Closed-form formulas are derived for the rank and inertia of submatrices of the Moore–Penrose inverse of a Hermitian matrix. A variety of consequences on the nonsingularity, nullity and definiteness of the submatrices are also presented.

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...