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

Sushobhan Maity (Department of Mathematics, Visva-Bharati, Santiniketan-731235, India)
A. K. Bhuniya (Department of Mathematics, Visva-Bharati, Santiniketan-731235, India)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 7, No 1 (2019): Electronic Journal of Graph Theory and Applications

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.