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Prove that the graphs G₁ and G₂ are isomorphic. f a e G₁ b d C 1 5 3 4 G₂ 2 6

Prove that the graphs G₁ and G₂ are isomorphic. f a e G₁ b d C 1 5 3 4 G₂ 2 6

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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Prove that graphs G1 and G2 are isomorphic. 

**Prove that the graphs \( G_1 \) and \( G_2 \) are isomorphic.** **Graph \( G_1 \):** - Vertices are labeled \( a, b, c, d, e, f \). - The structure is a hexagon where opposite vertices are connected by edges. - Specifically, vertex \( a \) connects to \( f, e, b, \) and \( c \). - Vertex \( b \) connects to \( a, c, e, \) and \( d \). - Vertex \( c \) connects to \( a, b, f, \) and \( d \). - Vertex \( d \) connects to \( c, b, e, \) and \( f \). - Vertex \( e \) connects to \( a, f, b, \) and \( d \). - Vertex \( f \) connects to \( a, e, c, \) and \( d \). **Graph \( G_2 \):** - Vertices are labeled \( 1, 2, 3, 4, 5, 6 \). - Structured as two connected triangles forming a square. - Specifically, vertex \( 1 \) connects to \( 2, 5, \) and \( 3 \). - Vertex \( 2 \) connects to \( 1, 3, 6, \) and \( 4 \). - Vertex \( 3 \) connects to \( 1, 2, 4, \) and \( 5 \). - Vertex \( 4 \) connects to \( 2, 6, 5, \) and \( 3 \). - Vertex \( 5 \) connects to \( 1, 3, 4, \) and \( 6 \). - Vertex \( 6 \) connects to \( 2, 4, 5, \) and \( 1 \). **Explanation:** Both graphs have 6 vertices and each vertex has 4 edges. To prove these graphs are isomorphic, find a one-to-one correspondence between vertices of \( G_1 \) and \( G_2 \) such that adjacency is preserved.
Transcribed Image Text:**Prove that the graphs \( G_1 \) and \( G_2 \) are isomorphic.** **Graph \( G_1 \):** - Vertices are labeled \( a, b, c, d, e, f \). - The structure is a hexagon where opposite vertices are connected by edges. - Specifically, vertex \( a \) connects to \( f, e, b, \) and \( c \). - Vertex \( b \) connects to \( a, c, e, \) and \( d \). - Vertex \( c \) connects to \( a, b, f, \) and \( d \). - Vertex \( d \) connects to \( c, b, e, \) and \( f \). - Vertex \( e \) connects to \( a, f, b, \) and \( d \). - Vertex \( f \) connects to \( a, e, c, \) and \( d \). **Graph \( G_2 \):** - Vertices are labeled \( 1, 2, 3, 4, 5, 6 \). - Structured as two connected triangles forming a square. - Specifically, vertex \( 1 \) connects to \( 2, 5, \) and \( 3 \). - Vertex \( 2 \) connects to \( 1, 3, 6, \) and \( 4 \). - Vertex \( 3 \) connects to \( 1, 2, 4, \) and \( 5 \). - Vertex \( 4 \) connects to \( 2, 6, 5, \) and \( 3 \). - Vertex \( 5 \) connects to \( 1, 3, 4, \) and \( 6 \). - Vertex \( 6 \) connects to \( 2, 4, 5, \) and \( 1 \). **Explanation:** Both graphs have 6 vertices and each vertex has 4 edges. To prove these graphs are isomorphic, find a one-to-one correspondence between vertices of \( G_1 \) and \( G_2 \) such that adjacency is preserved.
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