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Exploring Graph Classes and Their Properties: A Comprehensive Overview

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This resource delves into the various classes of graphs and their characteristics, highlighting statistical tools and MATLAB functions used for analysis. Key concepts include vertex degree histograms, average shortest path lengths, clustering coefficients, and Laplacian eigenvalues. Additionally, it categorizes graphs into types such as power-law, small-world, and planar graphs, while also introducing random graph generators. Various real-world graph examples, including finite element meshes and relationship networks, are discussed to illustrate the applicability and significance of graph theory in different domains.

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Exploring Graph Classes and Their Properties: A Comprehensive Overview

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Presentation Transcript

  1. CS240A: Classes of Graphsand Their Propertiesslides under construction – see the Matlab transcript for what I actually did in class

  2. Some graph statistics (and Matlab tools) • Vertex degree histogram: dhist.m • Avg shortest path length or BFS level profile: bfslevels.m • Clustering coefficient: ccoeff.m • c = 3*(# triangles) / (# connected triples) • Laplacian eigenvalues (and vectors): meshpart toolbox • Separator size: meshpart toolbox • Fill (chordal completion size): analyze.m and amd.m

  3. Some classes of graphs • Graphs observed in the wild (see Florida collection for many examples): • Finite element meshes: CGmats.m • Circuit simulation graphs: circuit_3.mat • Relationship networks: coAuthorsDBLP.mat, PGPgiantcompo.mat • … many others! • Classifications of graphs: • Power-law graphs • Small-world graphs • Planar graphs • Overlap graphs • Generators for classes of graphs: • Erdos-Renyi (flat) random graphs: sprandsym.m • RMAT random graph generator: rmat.m • 2-D and 3-D mesh generators: grid5.m etc. in meshpart toolbox

  4. RMAT Approximate Power-Law Graph

  5. Strongly Connected Components

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