On the spectrum of linear dependence graph of a finite dimensional vector space

Abstract

In this article, we introduce and characterize linear dependence graph Γ( V ) of a finite dimensional vector space V over a finite field of q elements. Two vector spaces U and V are isomorphic if and only if their linear dependence graphs Γ( U ) and Γ( V ) are isomorphic. The linear dependence graph Γ( V ) is Eulerian if and only if q is odd. Highly symmetric nature of Γ( V ) is reflected in its automorphism group S m ⊕ ( ⊕ i = 1 m S q − 1 ) , where m = ( q n − 1)/( q − 1) . Besides these basic characterizations of Γ( V ) , the main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph.