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The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only the latter ones here. Definition 1: A labeled multidigraph is a labeled graph with labeled arcs. Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs.
In mathematics, especially representation theory, a quiver is another name for a multidigraph; that is, a directed graph where loops and multiple arrows between two vertices are allowed. Quivers are commonly used in representation theory: a representation V of a quiver assigns a vector space V ( x ) to each vertex x of the quiver and a linear ...
A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
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).
These structures are generalization of the notion of accessible pointed graph (abbreviated as apg) from non-well-founded set theory. We will use the apq acronym for the below described multidigraph structures. This is meant as an abbreviation of "accessible pointed quiver" where quiver is an established synonym for "multidigraph".
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
A planar graph is a graph that has such an embedding onto the Euclidean plane, and a toroidal graph is a graph that has such an embedding onto a torus. The genus of a graph is the minimum possible genus of a two-dimensional manifold onto which it can be embedded.
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 no loops.