15-344/Classnotes for Thursday September 17

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Lecture Notes for September 17

DEFINITION 7 Isomorphism A graph G_1 = (V_1, E_1) is called isomorphic to a graph G_2 = (V_2, E_2) whenever there exists a bijection \phi:V_1 \rightarrow V_2 such that \forall a,b\in V_1 we have (ab)\in E_1 if and only if ( \phi (a) \phi (b))\in E_2. G_1\sim G_2 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: G \sim G A graph is isomorphic to itself

2. Symmetric: G_1 \sim G_2 \implies G_2 \sim G_1 In other words, for every \phi:V_1 \rightarrow V_2 we have \phi ^{-1} :V_2 \rightarrow V_1

3. Transitive: G_1 \sim G_2 , G_2 \sim G_3 \implies G_1 \sim G_3

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 G^c = (V,E^c)

  • Complement means (ab)\in E^c \Longleftrightarrow (ab)\notin E

DEFINITION 8 Subgraph A subgraph of a graph G = (V,E) is a graph G' = (V',E') such that V'\subset V and E'\subset E.

  • Checking if two graphs are isomorphic is a hard problem

Scanned Lecture Note for September 17

Scanned Tutorial Notes for September 17