A Study on some problems in Graph Theory

A problem in graph theory that has received increased attention during the past 50 years concerns studying methods of distinguishing the vertices of a connected graph from one another.There are many fields of mathematics now in the mathematics curriculum that overlap into graph theory. Large areas of set theory, pure combinatorics, algebra, geometry and especially topology consider problems of graph theory. However, since graph theory now makes its appearance in so many fields and especially since a large amount of graph theory could be developed and presented at the high school or undergraduate level, it would seem to merit more than just a passing glance in the curriculum. First, however, a great effort is needed to introduce graphs as a logical abstract mathematical system. A sequential rigorous development with preciseness of definitions and sufficiently complete to reveal its basic nature and applications is needed.