1. Lecture 13 Graphs

2. 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 All these questions can be answered using a mathematical object named a “graph” – what are graphs? – what are their basic properties?

3. Introduction to Graphs Graph – A set of vertices(nodes) V = {v 1, v 2, …., v n } – A set of edges(arcs) that connects the vertices E={e 1, e 2, …, e m } – Each edge e i 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)

4. 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 w 1, w 2, …, w n is a sequence of vertices such that (w i, w i+1 ) in E, then there is a path of length n-1 from w 1 to w n. 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 = {w 1, w 2, …, w n } is a sequence such that w 1 = w n and |S| >1 – A directed acyclic graph (DAG) is a directed graph having no cycles.

5. 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

6. 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

7. 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 C B F D E G A C B F D E G

8. Graph Algorithms Graphs depend on two parameters – edges (E) – Vertices (V) Graph algorithms can be complicated to analyze – One algorithm might be order (V 2 ) for dense graphs – Another might be order((E+V)log E) for sparse graphs 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.