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  2. Triangle-free graph - Wikipedia

    en.wikipedia.org/wiki/Triangle-free_graph

    In the mathematical area of graph theory, a triangle-free graph is an undirected graph in which no three vertices form a triangle of edges. Triangle-free graphs may be equivalently defined as graphs with clique number ≤ 2, graphs with girth ≥ 4, graphs with no induced 3-cycle, or locally independent graphs.

  3. Pseudoforest - Wikipedia

    en.wikipedia.org/wiki/Pseudoforest

    A 1-forest, sometimes called a maximal pseudoforest, is a pseudoforest to which no more edges can be added without causing some component of the graph to contain multiple cycles. If a pseudoforest contains a tree as one of its components, it cannot be a 1-forest, for one can add either an edge connecting two vertices within that tree, forming a ...

  4. Multiple edges - Wikipedia

    en.wikipedia.org/wiki/Multiple_edges

    Multiple edges joining two vertices. In graph theory, multiple edges (also called parallel edges or a multi-edge), are, in an undirected graph, two or more edges that are incident to the same two vertices, or in a directed graph, two or more edges with both the same tail vertex and the same head vertex. A simple graph has no multiple edges and ...

  5. Graph (discrete mathematics) - Wikipedia

    en.wikipedia.org/wiki/Graph_(discrete_mathematics)

    A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).

  6. Loop (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Loop_(graph_theory)

    In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same ...

  7. Pancyclic graph - Wikipedia

    en.wikipedia.org/wiki/Pancyclic_graph

    Mantel's theorem states that any -vertex undirected graph with at least / edges, and no multiple edges or self-loops, either contains a triangle or it is the complete bipartite graph /, /. This theorem can be strengthened: any undirected Hamiltonian graph with at least n 2 / 4 {\displaystyle n^{2}/4} edges is either pancyclic or K n / 2 , n / 2 ...

  8. Hypergraph - Wikipedia

    en.wikipedia.org/wiki/Hypergraph

    Empty - has no edges. Non-simple (or multiple) - has loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices. Simple - has no loops and no repeated edges.

  9. Connectivity (graph theory) - Wikipedia

    en.wikipedia.org/wiki/Connectivity_(graph_theory)

    This graph becomes disconnected when the right-most node in the gray area on the left is removed This graph becomes disconnected when the dashed edge is removed.. In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more ...