The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero o...

In this thesis, we make some contributions at the interface between ‘algebra’ and ‘graph theory’. In Chapter 1, we give an overview of the topics and also the definitions and preliminaries. In Chapter 2, we estimate the number of possible types degree patterns of k-lacunary polynomials of degree t < p which split completely modulo p. The result is based on a rather unusual combination of two te...

For a given simple graph G, S(G) is defined to be the set of real symmetric matrices A whose (i, j)th entry is nonzero whenever i 6= j and ij is an edge in G. In [2], ξ(G) is defined to be the maximum corank (i.e., nullity) among A ∈ S(G) having the Strong Arnold Property; ξ is used to study the minimum rank/maximum eigenvalue multiplicity problem for G. Since ξ is minor monotone, the graphs G ...

In this paper, we classify the irreducible integrable modules for nullity 2 toroidal extended affine Lie algebras with finite dimensional weight spaces.

Recently, attempts were made to generalize the undirected branching greedoid to a greedoid whose feasible sets consist of sets of edges containing the root satisfying additional size restrictions. Although this definition does not always result in a greedoid, the lift of the undirected branching greedoid has the properties desired by the authors. The k-th lift of a greedoid has sets whose nulli...

x∈Fpn en(f(x)), where en(y) = e2πiTrn(y)/p, y ∈ Fpn , Trn = TrFpn/Fp . There is an effective way to compute the nullity of the quadratic form Trmn(f(x)) for all integer m > 0. Assuming that all such nullities are known, we find relative formulas for S(f,mn) in terms of S(f, n) when νp(m) ≤ min{νp(αi) : 1 ≤ i ≤ k}, where νp is the p-adic order. We also find an explicit formula for S(f, n) when ν...

In this article we present a proof of the result known in Linear Algebra as the “rank nullity Theorem”, which states that, given any linear form f from a finite dimensional vector space V to a vector space W , then the dimension of V is equal to the dimension of the kernel of f (which is a subspace of V ) and the dimension of the range of f (which is a subspace of W ). The proof presented here ...

A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality involving the number of directed circuits in a circuit partition of a 2-in, 2-out digraph and the GF (2)-nullity of an associated matrix. This equality is essentially equivalent to the relationship between directed circuit partitions of 2-in, 2-out digraphs and vertexnullity interlace polynomials of circl...