Introduction To Graphs

Introduction: A graph G consists of a set of objects V = {v1, v2, v3, ......} called vertices (also called points or nodes) and other set E = {e1, e2, e3, .......} whose elements are called edges (also called lines or arcs). The set V(G) is called the vertex set of G and E(G) is the edge set. Usually the graph is denoted as G = (V, E)

Let G be a graph and {u, v} an edge of G. Since {u, v} is 2-element set, we may write {v, u} instead of {u, v}. It is often more convenient to represent this edge by uv or vu. If e = uv is an edge of a graph G, then we say that u and v are adjacent in G and that e joins u and v. (We may also say that each that of u and v is adjacent to or with the other).

For example : A graph G is defined by the sets
V(G) = {u, v, w, x, y, z} and E(G) = {uv, uw, wx, xy, xz}.

Now we have the following graph by considering these sets.

Every graph has a diagram associated with it. The vertex u and an edge e are incident with each other as are v and e. If two distinct edges say e and f are incident with a common vertex, then they are adjacent edges. A graph with p-vertices and q-edges is called a (p, q) graph. The (1, 0) graph is called trivial graph.

In the following figure the vertices a and b are adjacent but a and c are not. The edges x and y are adjacent but x and z are not. Although the edges x and z intersect in the diagram, their intersection is not a vertex of the graph.

Examples :
(1) Let V = {1, 2, 3, 4} and E = {{1, 2}, {1, 3}, {3, 2}. {4, 4}}. Then G(V, E) is a graph.
(2) Let V = {1, 2, 3, 4} and E = {{1, 5}, {2, 3}}. Then G(V, E) is not a graph, as 5 is not in V.

(3)

A graph with 5-vertices and 8-edges is called a (5, 8) graph.