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Show thet tegular a S(G) = A(G). G

Show thet tegular a S(G) = A(G). G

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 15** Show that a graph \( G \) is regular if and only if \( \delta(G) = \Delta(G) \). **Explanation:** - \( G \) regular: A graph is regular if every vertex has the same degree. - \( \delta(G) \): Minimum degree of the graph \( G \). - \( \Delta(G) \): Maximum degree of the graph \( G \). The problem asks to prove the equivalence between a graph \( G \) being regular and the condition that its minimum degree equals its maximum degree.
Transcribed Image Text:**Problem 15** Show that a graph \( G \) is regular if and only if \( \delta(G) = \Delta(G) \). **Explanation:** - \( G \) regular: A graph is regular if every vertex has the same degree. - \( \delta(G) \): Minimum degree of the graph \( G \). - \( \Delta(G) \): Maximum degree of the graph \( G \). The problem asks to prove the equivalence between a graph \( G \) being regular and the condition that its minimum degree equals its maximum degree.
Expert Solution
Step 1

Let us assume that graph G is regular.

Now, 

δ(G)= degree of the vertex with minimum degree 

∆G=degree of the vertex with maximum degree

Then by the definition of a regular graph, we see that "A graph is regular if all the vertices are of equal degree"

Hence, 

G is regular

⇒The degree of all the vertices of G are equal

Let the degree of all the vertices be "d"

Therefore this means that the minimum degree is "d" and the maximum degree is also "d"

Hence, δ(G)=∆(G)=d

Hence, if graph G is regular then δ(G)=∆(G)

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