On the nullity of some families of r-partite graphs

The nullity of a graph G, denoted by η(G) is defined to be the multiplicity of the eigenvalue zero in the spectrum of a graph. The spectrum of a graph G is a two-row matrix, the first row elements are the distinct eigenvalues of its adjacency matrix A(G) and the second row elements are its correspon...

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Main Author: Marcelino, Lyca DC
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Language:English
Published: Animo Repository 2020
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Online Access:https://animorepository.dlsu.edu.ph/etd_masteral/5958
https://animorepository.dlsu.edu.ph/context/etd_masteral/article/13016/viewcontent/Marcelino_Lyca_11790709_On_the_nullity_of_some_families_of_r_partite_graphs_Partial.pdf
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spelling oai:animorepository.dlsu.edu.ph:etd_masteral-130162022-05-05T01:45:49Z On the nullity of some families of r-partite graphs Marcelino, Lyca DC The nullity of a graph G, denoted by η(G) is defined to be the multiplicity of the eigenvalue zero in the spectrum of a graph. The spectrum of a graph G is a two-row matrix, the first row elements are the distinct eigenvalues of its adjacency matrix A(G) and the second row elements are its corresponding multiplicities. Furthermore, the rank of G, denoted by rank(G) is also the rank of A(G), that is rank(G) = rank(A(G)), which is defined as the maximum number of linearly independent row/column vectors in A(G). In addition, it is known that η(G) = n − rank(G), thus any result about rank can be stated in terms of nullity and vice versa. In this paper, we investigate three different families of r-partite graphs of order n and we determine the nullity of these r-partite families using its rank. First, a complete r-partite graphs denoted by Kn1,n2,n3,...,nr where n = n1 + n2 + n3 + ... + nr and r ≥ 4. Second, the family of r-partite graphs where n ≥ 2r − 1 and r ≥ 4 and is an extension of family of tripartite graphs introduced in the paper “On the nullity of a family of tripartite graphs” by Farooq, Malik, Pirzada and Naureen. While the third one is another family of r-partite graphs where n ≥ (2 r 2+r 2) and r ≥ 4. We characterize the third family with r-partition that satisfy properties different from what we obtain in the second family of r- partite graphs. 2020-10-03T07:00:00Z text application/pdf https://animorepository.dlsu.edu.ph/etd_masteral/5958 https://animorepository.dlsu.edu.ph/context/etd_masteral/article/13016/viewcontent/Marcelino_Lyca_11790709_On_the_nullity_of_some_families_of_r_partite_graphs_Partial.pdf Master's Theses English Animo Repository Graph theory Bipartite graphs Mathematics
institution De La Salle University
building De La Salle University Library
continent Asia
country Philippines
Philippines
content_provider De La Salle University Library
collection DLSU Institutional Repository
language English
topic Graph theory
Bipartite graphs
Mathematics
spellingShingle Graph theory
Bipartite graphs
Mathematics
Marcelino, Lyca DC
On the nullity of some families of r-partite graphs
description The nullity of a graph G, denoted by η(G) is defined to be the multiplicity of the eigenvalue zero in the spectrum of a graph. The spectrum of a graph G is a two-row matrix, the first row elements are the distinct eigenvalues of its adjacency matrix A(G) and the second row elements are its corresponding multiplicities. Furthermore, the rank of G, denoted by rank(G) is also the rank of A(G), that is rank(G) = rank(A(G)), which is defined as the maximum number of linearly independent row/column vectors in A(G). In addition, it is known that η(G) = n − rank(G), thus any result about rank can be stated in terms of nullity and vice versa. In this paper, we investigate three different families of r-partite graphs of order n and we determine the nullity of these r-partite families using its rank. First, a complete r-partite graphs denoted by Kn1,n2,n3,...,nr where n = n1 + n2 + n3 + ... + nr and r ≥ 4. Second, the family of r-partite graphs where n ≥ 2r − 1 and r ≥ 4 and is an extension of family of tripartite graphs introduced in the paper “On the nullity of a family of tripartite graphs” by Farooq, Malik, Pirzada and Naureen. While the third one is another family of r-partite graphs where n ≥ (2 r 2+r 2) and r ≥ 4. We characterize the third family with r-partition that satisfy properties different from what we obtain in the second family of r- partite graphs.
format text
author Marcelino, Lyca DC
author_facet Marcelino, Lyca DC
author_sort Marcelino, Lyca DC
title On the nullity of some families of r-partite graphs
title_short On the nullity of some families of r-partite graphs
title_full On the nullity of some families of r-partite graphs
title_fullStr On the nullity of some families of r-partite graphs
title_full_unstemmed On the nullity of some families of r-partite graphs
title_sort on the nullity of some families of r-partite graphs
publisher Animo Repository
publishDate 2020
url https://animorepository.dlsu.edu.ph/etd_masteral/5958
https://animorepository.dlsu.edu.ph/context/etd_masteral/article/13016/viewcontent/Marcelino_Lyca_11790709_On_the_nullity_of_some_families_of_r_partite_graphs_Partial.pdf
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