Question

a connected graph has 3 regions and 8 edges find number of vertices

Answer 1

$\underline{\textsf{Given:}}$

$\textsf{In a graph,}$

$\textsf{Number of edges is 8 and number of regions is 3}$

$\underline{\textsf{To find:}}$

$\textsf{Number of vertices}$

$\underline{\textsf{Solution:}}$

$\underline{\textsf{Concept used:}}$

$\bf\textsf{Euler's formula:}$

$\textsf{For any graph G,}$

$\boxed{\mathsf{V-E+F=2}}$

$\textsf{V-Number of vertices}$

$\textsf{E-Number of edges}$

$\textsf{F-Number of regions}$

$\textsf{Here, E=8 and F=3}$

$\textsf{By Eulers's formula}$

$\mathsf{V-E+F=2}$

$\implies\mathsf{V-8+3=2}$

$\implies\mathsf{V-5=2}$

$\implies\mathsf{V=5+2}$

$\implies\mathsf{V=7}$

$\underline{\textsf{Answer:}}$

$\textsf{Number of vertices is 7}$

Find more:

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?

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