Question

An undirected graph has twelve nodes. Four of them have degree six, five of them have degree three, three of them have degree seven. What is the number of edges in this graph?

Answer 1

$\text{Let G(p,q) be the given undirected graph}$

$\text{Here, p is the no. of vertices and q be the no. of edges}$

$\textbf{Given:}$

$p=12$

$\textbf{To find:}$

$\text{Number of edges of G}$

$\textbf{Solution:}$

$\text{Let v_1,v_2,........,v_{12} be vertices of G}$

$\text{As per given data,}$

$\text{Sum of the degrees of the vertices}$

$=\sum\limits_{i=0}^{i=12}\,d(v_i)$

$=(4{\times}6)+(5{\times}3)+(3{\times}7)$

$=24+15+21$

$=60$

$\text{But, we know that}$

$\sum\limits_{i=0}^{i=12}\,d(v_i)=2\,q$

$\implies\,60=2\,q$

$\implies\,q=\dfrac{60}{2}$

$\implies\bf\,q=30$

$\textbf{Answer:}$

$\text{Number of edges is 30}$