Lecture 009

Ada's Lecture

Undirected Graph (G = (V, E), where V, E are sets): may not be all reachable

Walk(path): walk in G is sequence of vertices v_0, v_1, v_2, ..., v_n such that (\forall 1 \leq t \leq n)(\{v_{t-1}, v_t\} \in E)

Cost Function: \lambda: E \rightarrow \mathbb{N}_+

Reachable (equivalence relation): v is reachable from u if there is a path from u to v

Reachable in B step as Boolean Matrix Multiplication: C(i, j) = \bigvee_k A(i, k) \land A(k, j)

Directed Graph: edge is ordered pair

General Graph: allow parallel edges and self-loops

Endpoints: u, v are endpoints of e = \{u, v\} \in E. Adjacent: u, v are adjacent when e = \{u, v\} \in E. Incident: u, v are incident on e = \{u, v\} \in E. Neighbore: u, v are neighbore of v, u respectively when e = \{u, v\} \in E. Neighborehood: neighborehood of vertice u is the set of vertices N(u) = \{v | \{u, v\} \in E\} connected to u. Degree: degree of u is |N(u)|

Adjacency Matrix: A[i,j] = \begin{cases} 1 \text{ if } i, j \text{ are adjacent}\\ 0 \text{ otherwise}\\ \end{cases}

Adjacency List: |V|-dimension array each storing its neighbores in chain.

Maximum Cut: separate vertice into two groups, making as many edges connected to different group as possible

Sutner's Lecture

Graph: equivalent to relation

Isomorphic: Let G_1 = \langle{V_1, E_1}\rangle, G_2 = \langle{V_2, E_2}\rangle. G_1, G_2 are isomorphic if there is a bijection f : V_1 \rightarrow V_2 such that (u, v) \in E_1 \iff (f(u), f(v)) \in E_2

Relationship Between Number of Edge and Vertices

Relationship Between Number of Edge and Vertices

Graph Data Structure

Graph Data Structure

Sparse Graph: |E| \leq |V|^2

Minor: graph H is a minor of G if it can be obtained from G by a sequence of vertex removals.

Planar Graph: can be drawned without any edges

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