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

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

<p>In this article, we introduce and characterize linear dependence graph <span class="math">Γ(<em>V</em>)</span> of a finite dimensional vector space <span class="math"><em>V</em></span> over a finite field of <span class=...

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Main Authors: Sushobhan Maity, A. K. Bhuniya
Format: Article
Language:English
Published: Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia 2019-04-01
Series:Electronic Journal of Graph Theory and Applications
Subjects:
graph, linear dependence, laplacian, distance, spectrum
Online Access:https://www.ejgta.org/index.php/ejgta/article/view/493
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spelling doaj-7eeb7b21c6c941d39f24f86cbbb5b4f02021-03-11T01:13:06ZengIndonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), IndonesiaElectronic Journal of Graph Theory and Applications2338-22872019-04-0171435910.5614/ejgta.2019.7.1.4135On the spectrum of linear dependence graph of a finite dimensional vector spaceSushobhan Maity0A. K. Bhuniya1Department of Mathematics, Visva-Bharati, Santiniketan-731235, IndiaDepartment of Mathematics, Visva-Bharati, Santiniketan-731235, India<p>In this article, we introduce and characterize linear dependence graph <span class="math">Γ(<em>V</em>)</span> of a finite dimensional vector space <span class="math"><em>V</em></span> over a finite field of <span class="math"><em>q</em></span> elements. Two vector spaces <span class="math"><em>U</em></span> and <span class="math"><em>V</em></span> are isomorphic if and only if their linear dependence graphs <span class="math">Γ(<em>U</em>)</span> and <span class="math">Γ(<em>V</em>)</span> are isomorphic. The linear dependence graph <span class="math">Γ(<em>V</em>)</span> is Eulerian if and only if <span class="math"><em>q</em></span> is odd. Highly symmetric nature of <span class="math">Γ(<em>V</em>)</span> is reflected in its automorphism group <span class="math"><em>S</em><sub><em>m</em></sub> ⊕ ( ⊕ <sub><em>i</em> = 1</sub><sup><em>m</em></sup><em>S</em><sub><em>q</em> − 1</sub>)</span>, where <span class="math"><em>m</em> = (<em>q</em><sup><em>n</em></sup> − 1)/(<em>q</em> − 1)</span>. Besides these basic characterizations of <span class="math">Γ(<em>V</em>)</span>, the main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph.</p>https://www.ejgta.org/index.php/ejgta/article/view/493graph, linear dependence, laplacian, distance, spectrum
collection DOAJ
language English
format Article
sources DOAJ
author Sushobhan Maity
A. K. Bhuniya
spellingShingle Sushobhan Maity
A. K. Bhuniya
On the spectrum of linear dependence graph of a finite dimensional vector space
Electronic Journal of Graph Theory and Applications
graph, linear dependence, laplacian, distance, spectrum
author_facet Sushobhan Maity
A. K. Bhuniya
author_sort Sushobhan Maity
title On the spectrum of linear dependence graph of a finite dimensional vector space
title_short On the spectrum of linear dependence graph of a finite dimensional vector space
title_full On the spectrum of linear dependence graph of a finite dimensional vector space
title_fullStr On the spectrum of linear dependence graph of a finite dimensional vector space
title_full_unstemmed On the spectrum of linear dependence graph of a finite dimensional vector space
title_sort on the spectrum of linear dependence graph of a finite dimensional vector space
publisher Indonesian Combinatorial Society (InaCombS); Graph Theory and Applications (GTA) Research Centre; University of Newcastle, Australia; Institut Teknologi Bandung (ITB), Indonesia
series Electronic Journal of Graph Theory and Applications
issn 2338-2287
publishDate 2019-04-01
description <p>In this article, we introduce and characterize linear dependence graph <span class="math">Γ(<em>V</em>)</span> of a finite dimensional vector space <span class="math"><em>V</em></span> over a finite field of <span class="math"><em>q</em></span> elements. Two vector spaces <span class="math"><em>U</em></span> and <span class="math"><em>V</em></span> are isomorphic if and only if their linear dependence graphs <span class="math">Γ(<em>U</em>)</span> and <span class="math">Γ(<em>V</em>)</span> are isomorphic. The linear dependence graph <span class="math">Γ(<em>V</em>)</span> is Eulerian if and only if <span class="math"><em>q</em></span> is odd. Highly symmetric nature of <span class="math">Γ(<em>V</em>)</span> is reflected in its automorphism group <span class="math"><em>S</em><sub><em>m</em></sub> ⊕ ( ⊕ <sub><em>i</em> = 1</sub><sup><em>m</em></sup><em>S</em><sub><em>q</em> − 1</sub>)</span>, where <span class="math"><em>m</em> = (<em>q</em><sup><em>n</em></sup> − 1)/(<em>q</em> − 1)</span>. Besides these basic characterizations of <span class="math">Γ(<em>V</em>)</span>, the main contribution of this article is to find eigen values of adjacency matrix, Laplacian matrix and distance matrix of this graph.</p>
topic graph, linear dependence, laplacian, distance, spectrum
url https://www.ejgta.org/index.php/ejgta/article/view/493
work_keys_str_mv AT sushobhanmaity onthespectrumoflineardependencegraphofafinitedimensionalvectorspace
AT akbhuniya onthespectrumoflineardependencegraphofafinitedimensionalvectorspace
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