Question

find an alternative 
definition of the degree of a vertex of a graph. 
please be detailed as possible thank you

Answer 1

Degree of a vertex of a graph:

Let G be a graph with vertex set V and edge set E.

Let and

The degree of a vertex is the number of edges incident on that vertex. The degree of vertex v_i is denoted as deg(v_i) or d(v_i).

Even Vertex and Odd Vertex:

If degree of a vertex is even then that vertex is called Even vertex.

If degree of a vertex is odd then that vertex is called Odd vertex.

Isolated Vertex:

If the degree of a vertex is zero then that vertex is called Isolated Vertex.

Pendent Vertex:

If the degree of a vertex is one then that vertex is called Pendent Vertex.

The minimum value of the degree of a vertex is zero and maximum it can be the 'number of vertices - 1'

We will see some examples where degree of every vertex is calculated.

1) Null graph :

Graph does not contain any edge. Thus no edge is incident on any vertex of a graph.

therefore degree of every vertex is given as:

d(v₁) = 0, d(v₂) = 0, d(v₃) = 0, d(v₄) = 0, d(v₅) = 0

All the vertices are even vertices because degree of every vertex is even (Zero) also all vertices are isolated vertices.

2) Graph G₂(V₂, E₂) :

e₁ edge is incident on a vertex P₁

deg(P₁) = 1

Similarly e₂ is incident on P₄deg(P₄) = 1

Therefore degree of every vertex is given as:

d(P₃) = 4, d(P₂) = 1, d(P₅) = 1

Vertices P₁, P₂, P₄ and P₅ are odd vertices because degree of those vertices is odd(1) also they are pendent vertices and P₃ vertex is even vertex.

3) Graph G₃(V₃, E₃) :

d(m₁) = 2, d(m₂) = 3, d(m₃) = 3, d(m₄) = 2, d(m₅) = 1 d(m₆) = 5

Odd Vertices : m₂, m₃,m₅ and m₆

Even Vertices : m₁, m₄

4) Complete Graph K₄:

d(n₁) = 3, d(n₂) = 3, d(n₃) = 3, d(n₄) = 3

All are odd vertices.

Also this is the example where degree of every vertex = Total number of vertices in graph -1

The minimum degree of a graph G is denoted by and maximum degree of a graph G is denoted by