Are the two graphs below equal? Are they isomorphic? If they are isomorphic, give the isomorphism. If not, explain. Graph 1: V = {a,b, c, d, e}, E = {{a,b}, {a,c},{a,e}, {b,d}, {b, e},{c,d}}. Graph 2: a d C
Are the two graphs below equal? Are they isomorphic? If they are isomorphic, give the isomorphism. If not, explain. Graph 1: V = {a,b, c, d, e}, E = {{a,b}, {a,c},{a,e}, {b,d}, {b, e},{c,d}}. Graph 2: a d C
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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**Graph Theory: Equality and Isomorphism**
**Question:**
Are the two graphs below equal? Are they isomorphic? If they are isomorphic, give the isomorphism. If not, explain.
**Graph 1:**
- **Vertices (V):** \{a, b, c, d, e\}
- **Edges (E):** \{\{a, b\}, \{a, c\}, \{a, e\}, \{b, d\}, \{b, e\}, \{c, d\}\}
**Graph 2:**
- **Diagram Description:**
- The graph is drawn with vertices labeled \(a, b, c, d, e\).
- Edges are as follows:
- \(a\) connects to \(b\) and \(e\).
- \(b\) connects to \(e\) and \(d\).
- \(c\) connects to \(d\) and \(a\).
**Analysis:**
- **Equality:**
- Two graphs are equal if they have the same set of vertices and edges. Here, the representation suggests different layouts, so an analysis of isomorphic properties is necessary to determine equality in structure.
- **Isomorphism:**
- To determine isomorphism, each vertex and its connecting edges in Graph 1 must correspond to a similar structure in Graph 2.
- By inspection, the mapping \(f(a) = e, f(b) = d, f(c) = c, f(d) = a, f(e) = b\) can form a proper correspondence between graphs if edge connectivity matches.
This mapping shows that Graph 1 and Graph 2 can be rearranged to be the same network structure through an isomorphism, which maintains structural properties like connectivity and vertex degree.
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