THE GRAPH THEORY IN MATHEMATICS-A STUDY

MATHEMATICS

A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense “related”. The objects of the graph correspond to vertices and the relations between them correspond to edges. A graph is depicted diagrammatically as a set of dots depicting vertices connected by lines or curves depicting edges. An edge E or ordered pair is a connection between two nodes u,v that is identified by unique pair (u,v). The pair (u,v) is ordered because (u,v) is not same as (v,u) in case of directed graph. The edge may have a weight or is set to one in case of unweighted graph. Undirected graph: A graph in which the direction of the edge is not defined. So if an edge exists between node ‘u’ and ‘v’, then there is a path from node ‘u’ to ‘v’ and vice versa. Connected graph: A graph is connected when there is a path between every pair of vertices. In a connected graph there is no unreachable node. Complete graph: A graph in which each pair of graph vertices is connected by an edge. In other words, every node ‘u’ is adjacent to every other node ‘v’ in graph ‘G’.A complete graph would have n(n-1)/2 edges. See below for proof. Biconnected graph: A connected graph which cannot be broken down into any further pieces by deletion of any vertex. It is a graph with no articulation point

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