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A graph with 6 vertices and 7 edges where the vertex number 6 on the far-left is a leaf vertex or a pendant vertex. In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph ...
between the edge space and the vertex space of . The incidence matrix of G {\displaystyle G} , as a linear transformation, maps each edge to its two incident vertices. Let v u {\displaystyle vu} be the edge between v {\displaystyle v} and u {\displaystyle u} then
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).
A vertex of an angle is the endpoint where two lines or rays come together. In geometry, a vertex (pl.: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. [1] [2] [3]
A vertex may exist in a graph and not belong to an edge. Under this definition, multiple edges , in which two or more edges connect the same vertices, are not allowed. Example of an undirected multigraph with 3 vertices, 3 edges and 4 loops.
Vertex contraction (sometimes called vertex identification) is similar, but the two vertices are not necessarily connected by an edge. Path contraction occurs upon the set of edges in a path that contract to form a single edge between the endpoints of the path. The inverse of edge contraction is vertex splitting. converse
remove_edge(G, x, y): removes the edge from the vertex x to the vertex y, if it is there; get_vertex_value(G, x): returns the value associated with the vertex x; set_vertex_value(G, x, v): sets the value associated with the vertex x to v. Structures that associate values to the edges usually also provide: [1] get_edge_value(G, x, y): returns ...
A graceful labeling; vertex labels are in black and edge labels in red. A graph is known as graceful if its vertices are labeled from 0 to | E |, the size of the graph, and if this vertex labeling induces an edge labeling from 1 to | E |. For any edge e, the label of e is the positive difference between the labels of the two vertices incident ...