Question

How many vertices a 4 regular graph with 10 edges has​

Answer 1

Answer:

How many vertices does a regular graph of degree 4 with 10 edges have?

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Let N be the total number of vertices.

According to Handshaking lemma:-

∑v ϵ Vdeg v=2|E|

Since degree of every vertices is 4, therefore sum of the degree of all vertices can be written as N×4

Put the value in above equation,

⟹N×4=2|E|

⟹N=2×104

⟹N=5 ■

Hence total vertices are 5 which signifies the pentagon nature of complete graph.

You can also visualise this by the help of this figure which shows complete regular graph of 5 vertices, :-

Answer 2

Given,

The number of graph$\[ = 4\]$

The number of edges$\[ = 10\]$

To find,

The number of vertices.

Solution,

Know that a graph with vertices V and m edges has the property, $\[2m = \sum\limits_{v \in V} {\deg \left( v \right).} \]$

Assume that the number of vertices is N.

Apply values.

$\[\begin{array}{l} \Rightarrow 2 \times 10 = N \times 4\\ \Rightarrow N = \frac{{2 \times 10}}{4}\\ \Rightarrow N = \frac{{20}}{4}\\ \Rightarrow N = 5\end{array}\]$

Hence, the number of vertices is $\[5\]$.