Relations between objects-

Data structures -

Graphs are discrete structures used in mathematics and computer science.

Road or computer networks-

Chemical bonds -

Stochastic processes-

Optimization (min-cut)-

Applications include the analysis or modeling of:

Example 10.1

Terminology.

Nodes are adjacent when they are endpoints of the same edge. An edge is incident upon its endpoints. The degree of a node is the number of incident edges.

A simple graph is a graph where no two edges connect the same vertices. A multigraph is a graph where an edge may connect the same vertices.

A directed graph has edges with associated direction. (Note: we

GraphsSaturday, November 25, 2017 7:19 PM

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A directed graph has edges with associated direction. (Note: we can no longer use the set to represent each edge … why? )

Example 10.2

The handshaking theorem.

Furthermore …

Try some …

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A simple graph is bipartite when its vertex set can be partitioned into two disjoint sets v1 and v2 such that every edge in the graph connects a vertex in v1 to a vertex in v2.

Example 10.3 Are the following graphs bipartite?

How can we determine whether a graph is bipartite?

Matching Problems.

Bipartite graphs are often used in matching problems, eg matching suppliers to consumers, workers to jobs, etc.

In this context a matching, is a subset of the edges M of some bipartite graph G = (V,E), where none of the edges in M are incident upon the same vertex.

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We say that a matching M in a bipartite graph G = (V,E) with bipartition (V1,V2) is a complete matching from V1 to V2 if every vertex in V1 is the endpoint of an edge in the matching, or equivalently, if |M| = |V1|.

Is there a complete matching here?

V1 V2

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Graph Representations.

Observe a graph edge set is a set of 2-tuples. It is a binary relation on the set of Vertices, thus it can be represented using a 0-1 matrix referred to as an adjacency matrix.

Paths on a graph.

Types of paths …

A path is a circuit or a cycle if it begins and ends with the same node.

An undirected graph is connected if there exists a path between all

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A path is a circuit or a cycle if it begins and ends with the same node.

An undirected graph is connected if there exists a path between all pairs of nodes.

A directed graph is strongly connected if there exists a path between all pairs of nodes.

A directed graph is weakly connected if for all pairs of nodes (vi,vj) there either exists a path from vi to vj or from vj to vi.

An attributed graph may have attributes associated with the vertices or the edges.

A weighted graph has weight attributes associated with each edge.

Shortest paths, Hamilton paths, and Euler paths

Path length in an unweighted graph, is the number of edges contained on the path.

Path length in a weighted graph is the sum of the weights of all edges on the path.

The shortest path on a Graph between a node A and a node B is a path from A to B with minimum length.

Example.

EULER PATHS

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Is there a faster way to determine whether an Euler Circuit/Path exists?

Hamilton Paths

Is there a simple way to find Hamilton Paths?

Find a Hamilton Circuit (if you can)

c p n

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Isomorphisms on a graph

Example from Rosen.

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