On graphs of minimum zero ring index

A ring R in which the product of any two elements is 0, where 0 is the additive identity of R, is called a zero ring. A new notion of vertex labeling for graphs, called zero ring labeling, is realized by assigning distinct elements of a zero ring to the vertices of the graph such that the sum of the...

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Main Author: Reynera, Michelle Dela Rosa
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Language:English
Published: Animo Repository 2018
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Online Access:https://animorepository.dlsu.edu.ph/etd_doctoral/552
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Institution: De La Salle University
Language: English
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spelling oai:animorepository.dlsu.edu.ph:etd_doctoral-15512021-05-18T10:40:03Z On graphs of minimum zero ring index Reynera, Michelle Dela Rosa A ring R in which the product of any two elements is 0, where 0 is the additive identity of R, is called a zero ring. A new notion of vertex labeling for graphs, called zero ring labeling, is realized by assigning distinct elements of a zero ring to the vertices of the graph such that the sum of the labels of adjacent vertices is not equal to the additive identity of the zero ring. The zero ring index of a graph G is the smallest positive integer (G) such that there exists a zero ring of order (G) for which G admits a zero ring labeling. Any zero ring labeling of G is optimal if it uses a zero ring consisting of (G) elements. Lower and upper bounds for (G) were determined, that is, n 6 (G) 6 2k, where n is the order of G and k is the value of the ceiling function of log2 n. In this study, families of graphs having zero ring indices attaining the lower bound are investigated. We obtained optimal zero ring labelings of common classes of graphs and presented an optimal zero ring labeling scheme for trees and cactus graphs. Additionally, we determined the zero ring index of graphs that result from graph operations such as join, Cartesian product, conjunction, composition and corona product, and established a relationship of the zero ring index of a graph obtained from a graph operation with the zero ring indices of the individual graphs. Necessary and sufficient conditions for graphs to have zero ring indices equal to their orders were also obtained. 2018-01-01T08:00:00Z text https://animorepository.dlsu.edu.ph/etd_doctoral/552 Dissertations English Animo Repository Graph labelings 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 labelings
Mathematics
spellingShingle Graph labelings
Mathematics
Reynera, Michelle Dela Rosa
On graphs of minimum zero ring index
description A ring R in which the product of any two elements is 0, where 0 is the additive identity of R, is called a zero ring. A new notion of vertex labeling for graphs, called zero ring labeling, is realized by assigning distinct elements of a zero ring to the vertices of the graph such that the sum of the labels of adjacent vertices is not equal to the additive identity of the zero ring. The zero ring index of a graph G is the smallest positive integer (G) such that there exists a zero ring of order (G) for which G admits a zero ring labeling. Any zero ring labeling of G is optimal if it uses a zero ring consisting of (G) elements. Lower and upper bounds for (G) were determined, that is, n 6 (G) 6 2k, where n is the order of G and k is the value of the ceiling function of log2 n. In this study, families of graphs having zero ring indices attaining the lower bound are investigated. We obtained optimal zero ring labelings of common classes of graphs and presented an optimal zero ring labeling scheme for trees and cactus graphs. Additionally, we determined the zero ring index of graphs that result from graph operations such as join, Cartesian product, conjunction, composition and corona product, and established a relationship of the zero ring index of a graph obtained from a graph operation with the zero ring indices of the individual graphs. Necessary and sufficient conditions for graphs to have zero ring indices equal to their orders were also obtained.
format text
author Reynera, Michelle Dela Rosa
author_facet Reynera, Michelle Dela Rosa
author_sort Reynera, Michelle Dela Rosa
title On graphs of minimum zero ring index
title_short On graphs of minimum zero ring index
title_full On graphs of minimum zero ring index
title_fullStr On graphs of minimum zero ring index
title_full_unstemmed On graphs of minimum zero ring index
title_sort on graphs of minimum zero ring index
publisher Animo Repository
publishDate 2018
url https://animorepository.dlsu.edu.ph/etd_doctoral/552
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