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### Lecture 13

Graphs

Introduction to Graphs

- Examples of Graphs
- Airline Route Map
- What is the fastest way to get from Pittsburgh to St Louis?
- What is the cheapest way to get from Pittsburgh to St Louis?

- Electric Circuits
- Circuit elements - transistors, resistors, capacitors
- is everything connected together?
- Depends on interconnections (wires)

- If this circuit is built will it work?
- Depends on wires and objects they connect.

- Job Scheduling
- Interconnections indicate which jobs to be performed before others
- When should each task be performed

- Airline Route Map
- All these questions can be answered using a mathematical object named a “graph”
- what are graphs?
- what are their basic properties?

Introduction to Graphs

- Graph
- A set of vertices(nodes) V = {v1, v2, …., vn}
- A set of edges(arcs) that connects the vertices E={e1, e2, …, em}
- Each edge ei is a pair (v, w) where v, w in V
- |V| = number of vertices (cardinality)
- |E| = number of edges

- Graphs can be
- directed (order (v,w) matters)
- Undirected (order of (v,w) doesn’t matter)

- Edges can be
- weighted (cost associated with the edge)
- eg: Neural Network, airline route map(vanguard airlines)

Paths and Graphs

- A path is a sequence of edges from one node to another
- A length of a path is the number of edges
- If w1, w2, …, wn is a sequence of vertices such that (wi, wi+1) in E, then there is a path of length n-1 from w1 to wn.
- what are paths of length 2 in vanguard airlines map?

- A simple path is a path where no vertex is repeated
- first and last vertices can be the same
- what is an example of a simple path in the vanguard map?
- What is an example of a non-simple path?

- A cycle (in a directed graph) is a path that begins and ends in the same vertex.
- i.e. S = {w1, w2, …, wn} is a sequence such that w1 = wn and |S| >1
- A directed acyclic graph (DAG) is a directed graph having no cycles.

More Graph Definitions

- Connected Graph
- A undirected graph is connected if for all (u,v) in V, there exists a path from u to v.
- A directed graph is strongly connected if for all (u,v) in V, there exists a path from u to v.
- A directed graph is weakly connected if for all (u,v) in V, either (u,v) is in E or (v,u) is in E.

- Complete Graph
- A graph with all nodes connected to each other directly

- Maximal Edge count
- undirected graph |E|max = (n-1) + (n-2) + … + 2 + 1 = n(n-1)/2
- directed graph |E|max = n(n-1)

- Degree
- number of edges incident to a vertex
- in a directed graph
- in-degree(sink) - number of edges into vertex
- out-degree(source) - number of edges from vertex

Some Cardinality Relations

- For most graphs |E| |V|2
- A dense graph when most edges are present
- E = (|V|2)
- A dense graph
- large number of edges (quadratic)
- Also |E| > |V| log |V| can be considered dense

- A sparse graph is a graph with relatively few edges
- no clear definition
- metric for sparsity |E| < |V| log |V|
- eg: ring computer network

More About Graphs

- A graph with no cycles is called a tree.
- This is a general definition of a tree
- A group of disconnected trees is called a “forest”
- A spanning tree of a graph is a subgraph that contains all vertices but only enough of the edges to form a tree.
- Any additional edge to a tree will form a cycle
- A tree with V vertices has exactly V-1 edges (induction proof)

A

A

C

G

C

G

B

B

D

D

E

E

F

F

Graph Algorithms

- Graphs depend on two parameters
- edges (E)
- Vertices (V)

- Graph algorithms can be complicated to analyze
- One algorithm might be order (V2)
- for dense graphs

- Another might be order((E+V)log E)
- for sparse graphs

- One algorithm might be order (V2)
- Depth First Search
- Is the graph connected? If not what are their connected components?
- Does the graph have a cycle?
- How do we examine (visit) every node and every edge systematically?
- First select a vertex, set all vertices connected to that vertex to non-zero
- Find a vertex that has not been visited and repeat the step above
- Repeat above until all zero vertices are examined.

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