Let G be a graph with n(G) vertices and m(G) be its matching number. The nullity of G, denoted by η(G), is the multiplicity of the eigenvalue zero of adjacency matrix of G. It is well known that if G is a tree, then η(G) = n(G)− 2m(G). Guo et al. [Jiming GUO, Weigen YAN, Yeongnan YEH. On the nullity and the matching number of unicyclic graphs. Linear Alg. Appl., 2009, 431: 1293–1301] proved tha...

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...

We present a type system for parametric object mutability and reference nullity in an imperative object oriented language. We present a simple but powerful system for generic nullity constraints, and build on previous work to provide safe initialization of objects which is not bound to constructors. The system is expressive enough to handle initialization of cyclic immutable data structures, an...

We introduce magic measures to quantify the nonstabilizerness of multiqubit quantum gates and establish lower bounds on $T$ count for fault-tolerant computation. First, we stabilizer nullity multi-qubit unitary, which is based subgroup quotient Pauli group associated with unitary. This unitary extends state-stabilizer by Beverland et al. a dynamic version. In particular, show this measure has d...