On the spectrum of linear dependence graph of a finite dimensional vector space - Sushobhan Maity; A. K. Bhuniya (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2019)
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 Sm ⊕ ( ⊕ i = 1mSq − 1), where m = (qn − 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.