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6.1: Graph Theory

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    There are several definitions that are important to understand before delving into Graph Theory. They are:

    • A graph is a picture of dots called vertices and lines called edges.
    • An edge that starts and ends at the same vertex is called a loop.
    • If there are two or more edges directly connecting the same two vertices, then these edges are called multiple edges.
    • If there is a way to get from one vertex of a graph to all the other vertices of the graph, then the graph is connected.
    • If there is even one vertex of a graph that cannot be reached from every other vertex, then the graph is disconnected.

    Example \(\PageIndex{1}\): Graph Example

    Figure \(\PageIndex{1}\): Graph 1

    In the above graph, the vertices are U, V, W, and Z and the edges are UV, VV, VW, UW, WZ1, and WZ2.

    This is a connected graph. VV is a loop. WZ1, and WZ2 are multiple edges.

    Example \(\PageIndex{2}\): Graph Example

    Figure \(\PageIndex{2}\): Graph 2 Figure \(\PageIndex{3}\): Graph 3

    The graph in Figure \(\PageIndex{2}\) is connected while the graph in Figure \(\PageIndex{3}\) is disconnected.

    Definition: Graph Concepts and Terminology

    • Order of a Network: the number of vertices in the entire network or graph
    • Adjacent Vertices: two vertices that are connected by an edge
    • Adjacent Edges: two edges that share a common vertex
    • Degree of a Vertex: the number of edges at that vertex
    • Path: a sequence of vertices with each vertex adjacent to the next one that starts and ends at different vertices and travels over any edge only once
    • Circuit: a path that starts and ends at the same vertex
    • Bridge: an edge such that if it were removed from a connected graph, the graph would become disconnected

    Example \(\PageIndex{3}\): Graph Terminology

    Figure \(\PageIndex{4}\): Graph 4

    In the above graph the following is true:

    • Vertex A is adjacent to vertex B, vertex C, vertex D, and vertex E.
    • Vertex F is adjacent to vertex C, and vertex D.
    • Edge DF is adjacent to edge BD, edge AD, edge CF, and edge DE.

    The degrees of the vertices:

    A 4
    B 4
    C 4
    D 4
    E 4
    F 2

    Here are some paths in the above graph: (there are many more than listed)




    Here are some circuits in the above graph: (there are many more than listed)



    F,C, E, D, F

    The above graph does not have any bridges.

    This page titled 6.1: Graph Theory is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Maxie Inigo, Jennifer Jameson, Kathryn Kozak, Maya Lanzetta, & Kim Sonier via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.