#### Title

### On graphs of minimum zero ring index

#### Date of Publication

2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Subject Categories

Mathematics

#### College

College of Science

#### Department/Unit

Mathematics and Statistics

#### Thesis Adviser

Leonor A. Ruivivar

#### Defense Panel Chair

Yvette F. Lim

#### Defense Panel Member

Francis Joseph H. Campena

Isagani B. Jose

Severino B. Gervacio

Neil M. Mame

#### Abstract/Summary

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.

#### Abstract Format

html

#### Language

English

#### Format

Electronic

#### Accession Number

CDTG007627

#### Shelf Location

Archives, The Learning Commons, 12F Henry Sy Sr. Hall

#### Physical Description

1 computer disc; 4 3/4 in.

#### Keywords

Graph labelings

#### Recommended Citation

Reynera, M. D. (2018). On graphs of minimum zero ring index. Retrieved from https://animorepository.dlsu.edu.ph/etd_doctoral/552