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.

• 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: $G\sim G$ 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

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  Answers of sample question