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

Sushobhan Maity, A. K. Bhuniya

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 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.


Keywords


graph, linear dependence, Laplacian, distance, spectrum

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DOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.4

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ISSN: 2338-2287

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