15-344/Classnotes for Thursday September 17
Lecture Notes for September 17
DEFINITION 7 Isomorphism A graph is called isomorphic to a graph whenever there exists a bijection such that we have if and only if . means they are isomorphic to each other.
- A bijection is a one-to-one and on-to function. https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection
- Isomorphism does not mean two things are identical but means they are mathematically the same.
The relationship of isomorphisms:
1. Reflexive: A graph is isomorphic to itself
2. Symmetric: In other words, for every we have
CLAIM If two graphs are isomorphic, then they have:
1. same number of vertices
2. same number of edges
3. vertex degrees (valencies) are the same between the two. For example, if one graph has 3 vertices of degree 2, and 2 vertices of degree 1, then the other graph should have the same
4. same number of subgraphs
5. same number of complements denoted by
- Complement means
DEFINITION 8 Subgraph A subgraph of a graph is a graph such that and .
- Checking if two graphs are isomorphic is a hard problem