Question

state and prove the fundamental theorem of graph theory.​

Answer 1

Concept:

Graph is used to describe the relationship betweedn lines and points. The axis of a graph is divided into two parts, $x-$axis and $y-$axis.

Given:

Graph theory.

Find:

State and prove the fundamental theorem of graph theory.​

Solution:

According to the problem,

The fundamental theorem of graph theory states that the sum of degree of all vertices is equal to twice the number of edges.

Proof:

Let $G$ be any graph with $e$ edges and $v$ vertices such as $v_{1},v_{2},v_{3},.........v_{n}$

Since each edge contributes two degrees,

Therefore, the sum of the degrees of all verices is the twice number of edges in $G$.

Sum of degree$=2[Edges]$

Thus it can be written as, $\sum_{1}d(v_{i})=2e$

Hence the proof fundamental theorem of graph theory that states that sum of vertices is equal to twice the number of edges is $\sum_{1}d(v_{i})=2e$