15-344/Classnotes for Thursday September 17

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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.

  • 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

3. Transitive:

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

Scanned Lecture Note for September 17

Scanned Tutorial Notes for September 17