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5: Graph Theory

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    88867

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    • 5.1: Discovering Graphs
      This page explains how to define graphs through various properties, using examples and checkpoints to distinguish valid graph formations from invalid ones. It employs set notation and prompts readers to analyze necessary and forbidden properties of graphs, facilitating a translation between set forms and diagrams.
    • 5.2: Properties of Graphs
      This page provides definitions and examples of graph properties like adjacency, vertex degrees, and types of graphs (regular, complete, bipartite). It covers subgraphs, graph complements, and duals, along with practice checkpoints for calculating degrees and understanding independent sets and maximum matchings. Each definition is illustrated with examples to aid in the comprehension of graph theory concepts.
    • 5.3: Graph Isomorphism
      This page explores graph isomorphism, which involves establishing a bijection between the vertex sets of two graphs while preserving their edge connections. It includes an example for clarification and challenges readers to identify isomorphic graphs from provided figures, along with the isomorphism function.
    • 5.4: Paths
      This page discusses key concepts in graph theory, including definitions of walks, trails, and paths, highlighting their distinct characteristics regarding vertex and edge repetition. It explains Eulerian trails and Hamiltonian paths and addresses graph connectivity, stating that a graph is connected if any vertex pair has a connecting path. Additionally, it introduces \(n\)-connected graphs and includes practice checkpoints to reinforce understanding of these concepts.
    • 5.5: Cycles
      This page defines important graph theory terms such as circuit, cycle, Eulerian circuit, and Hamiltonian circuit. It explains that a circuit is a closed walk with the same starting and ending vertex, while a cycle does not repeat vertices. An Eulerian circuit covers every edge once, and a Hamiltonian circuit visits each vertex once. Additionally, it offers practice checkpoints for readers to engage with the concepts, including drawing cycles and identifying specific types of circuits.
    • 5.6: Trees
      This page defines key concepts in graph theory, specifically trees, leaves, and spanning trees. It clarifies that a tree is a connected graph without cycles, a leaf is a vertex with a degree of one, and a spanning tree includes all vertices of a graph while maintaining tree properties. The content includes practice checkpoints for identifying trees, drawing non-isomorphic trees, proving tree properties, and finding spanning trees in various graphs.
    • 5.7: Linear Recurrence Relations
      This page explores recursive definitions in mathematics, focusing on continued radicals, continued fractions, and Fibonacci sequence. It covers recurrence relations and counting combinations while avoiding consecutive characters in strings. The text includes practice problems on food combinations and counting steps, highlighting the effectiveness of recursive strategies in problem-solving.

    This page titled 5: Graph Theory is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Mark A. Fitch via source content that was edited to the style and standards of the LibreTexts platform.