UNIT 1 BASIC PROPERTIES OF GRAPH Q1. What is a Graph? Ans . A graph is a coll ect ion of poi nts ca lle d ‘ver tic es’ and a colle cti on of lines cal led ‘edges’, each of which joins either a pair of points or a single point to itself. Each point is called a vertex, and the curve joining any pair is called an edge in a graph. Forexample - a road map which shows towns as vertices and the roads joining them as edges. Mathematically for every edge e of a graph we define a set { v1 , v2} of vertices which specifies that e joins vertices v1 and v2, where of course we need to allow the possibility that v1 = v2. Now this set {v1 , v2}, which we denote by δ(e), is a subset of the set of vertices. Therefore δ(e) is an elementof the power set of the vertex set. An undirected graph Gcomprises: (i) a finite non-empty set Vofvertices , (ii) a finite setEofedges, and (iii) a function δ :E→ (V ) such that, for every edge e, δ(e) is a one- or two-element subset ofV. The edge e is said tojoin the element(s) ofδ(e). Clearly Graph G has vertex set {v1 , v2 , v3 , v4} and edges set {e1 , e2 , e3 , e4 , e5}. The function δ :E→ (v) is defined by δ : e1 _→ {v1} δ : e2 _→ {v1 , v2} δ : e3 _→ {v1 , v3} δ : e4 _→ {v2 , v3} δ : e5 _→ {v2 , v3}. This simply indicates that e1 joins vertex v1 to itself, e2 joins vertices v1 and v2, etc. We emphasize that an edge may join a vertex to itself, as in the case ofe1, and a vertex may be connected to no edges at all, as in the case ofv4. Also note that a given pair of vertices may be joined by more than one edge; in this example the edges e4 and e5 both connect the vertices v2 and v3.
