A Study on Circular Distances in Graphs

Abstract

The present thesis discuss the circular distance and circular D-distance between vertices of a graph. The circular distance was first introduced by V. Kaladevi et al. in 2012. In chapter 1, Preliminaries, organization of thesis was discussed. Some basic definitions were collected. In chapter 2, we studied the circular distance between vertices of a graph. It has been proved that circular distance is a metric on the set of all vertices of graph. newlineIn chapter 3, some new graphs have been constructed which are called circular antipodal graphs. We looked at the circular antipodal graph of trees. In a tree all circular antipodal vertices are pendent vertices. But all pendent vertices need not be circular antipodal. In chapter 4, using circular distance, we constructed circular distant divisor graphs. We studied the construction of circular distant divisor graphs of double star graph. Further we constructed the corona graph of circular distant divisor graphs. newlineIn Chapter 5, we introduced the circular D-distance. In addition to the circular distance, we consider the degree of each vertex in the circular path. We computed the circular D-radius and circular D-diameter of some classes of graphs. In chapter 6, we found the average circular D-distance and in addition to that, we studied the circular Wiener index and circular D-wiener index of a graph. We carried out a relation between circular Wiener index and circular D-Wiener index. newlineIn chapter 7, we consider the double vertex graph of a given graph and studied the geodesic distance between vertices. Some properties have been discussed and computed the radius and diameter of some classes of graphs. newline newline