Graph Theory

A graph G = (V,E) consists of two sets V and E. The elements of V are called vertices while that of E are called edges. Each edge e is associated with an unordered pair vi,vj of vertices

An edge having the same vertex at both ends is called a loop.

Two edges associated with the same pair of end vertices are called parallel edges.

A graph which does not contain loops or parallel edges is called a simple graph.

The number of edges incident on a vertex v with loops being counted twice is called the degree of the vertex v.

A graph in which all vertices are of equal degree is called a regular graph.

A vertex having no incident edge is called an isolated vertex.

A graph having no edges is called a null graph.

Theorem 1 :

The sum of the degrees of all vertices of a graph is equal to twice the number of edges of the graph.

Theorem 2 :

A graph always contains a even number of vertices of odd degree.

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