### 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.

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

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