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Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not.

Is it possible for two different (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not.

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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**Question:** Is it possible for two *different* (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not. **Explanation:** Yes, it is possible for two non-isomorphic graphs to have the same number of vertices and the same number of edges. Additionally, two graphs can also have the same degree sequence, meaning the degrees of their vertices are the same, yet the graphs remain non-isomorphic. To understand this concept, we can consider the following example: Suppose we have two graphs, Graph A and Graph B, each consisting of five vertices with vertex degrees of 1, 2, 2, 3, and 4. These graphs are non-isomorphic because, although they have the same number of vertices and edges as well as the same degree sequence, their structures differ in terms of how these vertices are connected. While it is possible to sketch these graphs, the key point is recognizing that isomorphic graphs are structurally identical, meaning there exists a one-to-one correspondence between their vertex sets that preserves adjacency. Hence, two graphs sharing the same basic numerical features can nonetheless be arranged differently in terms of their topology.
Transcribed Image Text:**Question:** Is it possible for two *different* (non-isomorphic) graphs to have the same number of vertices and the same number of edges? What if the degrees of the vertices in the two graphs are the same (so both graphs have vertices with degrees 1, 2, 2, 3, and 4, for example)? Draw two such graphs or explain why not. **Explanation:** Yes, it is possible for two non-isomorphic graphs to have the same number of vertices and the same number of edges. Additionally, two graphs can also have the same degree sequence, meaning the degrees of their vertices are the same, yet the graphs remain non-isomorphic. To understand this concept, we can consider the following example: Suppose we have two graphs, Graph A and Graph B, each consisting of five vertices with vertex degrees of 1, 2, 2, 3, and 4. These graphs are non-isomorphic because, although they have the same number of vertices and edges as well as the same degree sequence, their structures differ in terms of how these vertices are connected. While it is possible to sketch these graphs, the key point is recognizing that isomorphic graphs are structurally identical, meaning there exists a one-to-one correspondence between their vertex sets that preserves adjacency. Hence, two graphs sharing the same basic numerical features can nonetheless be arranged differently in terms of their topology.
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