CSE 222
Systems Programming
Graph Theory Basics
Dr. Jim Holten
Introduction
• Basic Definitions
• Concepts and Algorithms
• Applications
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Basic Definitions
• Nodes aka Vertices
• Links aka Edges
aka Arcs
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Basic Definitions
• Subgraphs
• Undirected vs Directed Graphs
• Paths, Distances, and Graph Diameter
• Connected Graphs
• Components of a Graph
• Fully connected graph or subgraph
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Undirected Graph
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Directed Graph
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Basic Definitions
• Node Attributes aka Labels, Indices,
Weights, Categories, etc.
• Link Attributes aka Labels, Indices,
Weights, Categories, etc.
• A Semantic Graph is a graph partitioned
based on node or link attribute values
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Semantic Graph
Authors
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Documents
Concepts
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More Definitions
• A Hypernode replaces a collection of
nodes and their included links.
• A Hypergraph is any graph that includes
one or more hypernodes.
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Example of Hypergraph
Authors
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Documents
Concepts
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Hypergraph of Social Network
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Some Concepts
• A cluster is a subgraph made up of set of
highly interconnected nodes.
• A critical node is a node on a unique path
between two nodes or subgraphs of
interest.
• A critical link is a link on a unique path
between two nodes or subgraphs of
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Related Algorithms
• Finding clusters is defined many ways
– Fully connected subgraph or just better than
some threshold on connectedness?
– Is a node only connected to a node in the
cluster to be included in the cluster?
• Finding critical paths, nodes, or links
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Related Concepts
• Projections of nodes or subgraphs are
mappings via directed links.
• The domain subgraph includes the projection's
"from" nodes.
• The range subgraph includes the projection's
"to" nodes.
• Basis Node Set is a minimal set of nodes that
have a path to every other node
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Projection Mapping
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Subset Projections
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Related Concepts
• There are multiple ways of looking at a
single collection of nodes and links.
• Are multiple graphs just a single multicomponent graph?
• Interdependencies among multiple graphs
can be useful.
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Interdependencies
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Applications
• Software Organization
• Activity Sequence Organization
• Interactions between processes
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Software Organization
• Flow charts
• Data interdependencies
• Interactions between processes
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Data Organization Chart
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Processes Organization Chart
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Activity Sequence Organization
• Gantt and Pert charts
• Project resource planning charts
• Storyboarding
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Storyboard of the
Software Development Process
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Network Modeling Applications
• Communications (Internet, etc.)
• Energy distribution (electric power, natural gas,
etc.)
• Transportation (highways, railways, waterways,
etc)
• Social associations
• Financial and product flows
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Infrastructure Networks
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Critical Infrastructure
Interdependencies
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Physics Models
• Mesh Models
– Space segmented into volumes
– Volumes surrounded by faces and edges
– Edges meet at vertices
• Model element type dependencies form a
graph of interdependent element sets
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Mesh Models
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Element Type Dependencies
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Summary
Graphs are a significant factor in the present
and the future of computer science!
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