Let G = (V, E) be a bipartite graph with partite sets X and Y. Suppose X = {x₁,x2,...,n} and Y = {9₁, 92,..., ym}. Show that Σdg(x₁) = |E|.
Let G = (V, E) be a bipartite graph with partite sets X and Y. Suppose X = {x₁,x2,...,n} and Y = {9₁, 92,..., ym}. Show that Σdg(x₁) = |E|.
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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Question
![5. Let \( G = (V, E) \) be a bipartite graph with partite sets \( X \) and \( Y \). Suppose \( X = \{x_1, x_2, \ldots, x_n\} \) and \( Y = \{y_1, y_2, \ldots, y_m\} \). Show that
\[
\sum_{i=1}^{n} d_G(x_i) = |E|.
\]
(Hint: Think graph fairy). (Also, note that \(\sum_{j=1}^{m} d_G(y_j) = |E|\) as well, but just prove the above).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9123907a-8746-4d91-90b8-859e0a45c1f5%2Fd57e645c-f8c1-4b64-9da6-7e2f149355ff%2F1i1kakm_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let \( G = (V, E) \) be a bipartite graph with partite sets \( X \) and \( Y \). Suppose \( X = \{x_1, x_2, \ldots, x_n\} \) and \( Y = \{y_1, y_2, \ldots, y_m\} \). Show that
\[
\sum_{i=1}^{n} d_G(x_i) = |E|.
\]
(Hint: Think graph fairy). (Also, note that \(\sum_{j=1}^{m} d_G(y_j) = |E|\) as well, but just prove the above).
Expert Solution
Step 1
Let |E|=e.
Claim: .
Since G is bipartite with bipartite set X and Y.
Each edge in E(G) will contribute to the degree of two different vertices.
Therefore, the sum of the degrees should be exactly two times the number of edges.
So .
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