The nonzero component graph of near-vector space

Abstract

In this paper, we introduce a nonzero component graph of a finite-dimensional near-vector space constructed from a finite field. We investigate the relationship between the existence of a basis of a near-vector space over a field with that of the vector space over the same field, with the same dimension. We introduce the notion of a coordinate vector and prove an equivalent formulation for the definition of the nonzero component graph. We show that despite the vast differences in the structure of a near-vector space compared with a vector space, especially in the nonregular case, the graph fails the recognition problem in that it cannot distinguish between the near-vector space and its associated vector space. Thus, we show that the nonzero component graph of a near-vector space is identical to the one for a vector space over the same field, of the same dimension. Leveraging this result, we completely characterise and describe graph properties for the nonzero component graph on both near-vector spaces and vector spaces by partitioning the graph into components using the regularity graph of any nonregular near-vector space (with the same dimension) with its nonzero component graph. In closing we provide a reconstruction algorithm for constructing a near-vector space for a given nonzero component graph, satisfying certain conditions.