# On the spectrum of linear dependence graph of finite dimensional vector spaces

## TL;DR

This paper introduces the linear dependence graph of finite dimensional vector spaces over finite fields, exploring its properties and spectral characteristics, and establishing a link between graph isomorphism and vector space isomorphism.

## Contribution

It presents a novel graph structure for vector spaces and analyzes its properties and spectra, connecting graph isomorphism with vector space isomorphism.

## Key findings

- Linear dependence graphs are connected and sometimes complete.
- Isomorphic vector spaces have isomorphic linear dependence graphs.
- Spectral properties of the graphs are characterized and studied.

## Abstract

In this paper, we introduce a graph structure called linear dependence graph of a finite dimensional vector space over a finite field. Some basic properties of the graph like connectedness, completeness, planarity, clique number, chromatic number etc. have been studied. It is shown that two vector spaces are isomorphic if and only if their corresponding linear dependence graphs are isomorphic. Also adjacency spectrum, Laplacian spectrum and distance spectrum of the linear dependence graph have been studied.