Geometric Dual of Graphs and Related Aspects

Abstract

The main focus of this thesis is on the geometric dual of planar graphs. The problems newlineconsidered include * isomorphism of graphs, Haary graphs, SD graphs, HB graphs, newlineR-complements of planar graphs and the related aspects. We investigate the necessary newlineand sufficient conditions for two graphs to be *isomorphic. newlineWe define the boundary matrix and region matrix of planar graphs and study the newlineinterrelated results. One of the main results on *isomorphism of graphs is included in the newlinethesis, is an algorithm for *isomorphic graphs. newlineWe define the different types of totally disconnected graphs and study the perspective of newlinetheir geometric dual. We build the bridge between these graphs and * isomorphic graphs. newlineThere is a correspondence between trees and the geometric dual of Haary graphs, which newlineis explained meticulously, in this thesis. newlineWith the help of the geometric dual of graphs, we describe SD graphs with proper newlineexamples and prove the correlated results. By considering the definition of an SD graph, newlinethe first geometric number and the greatest geometric number come into the center of newlineattention. We prove formulae for geometric numbers of graphs and find the number of newlinegraphs with distinct geometric numbers on n vertices. newlineIn view of the geometric dual of graphs, we define Rank graphs and a Regular Rank newlinegraph and elucidate the related results with suitable examples. It is observed that two rank newlinegraphs are * isomorphic if and only if they have an equal number of regions of the same newlinerank. newlineWe enlighten the pioneering and an important concept of planar graphs, known as HB newlinegraphs. The pivot region number (PRN) of every planar graph is defined and proves that newlinethe PRN of any planar graph is at most four. By using the similar logic, we give the newlineminiature proof of the famous four color problem. newlineWe describe the special concept of coloring of graphs called as the perfect coloring of newlinegraphs and state the conjecture with partial proof by considering particular types of newlinegraphs. We correlate this coloring to the semi perfect coloring given b