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In graph theory, a vertex is incident with an edge if the vertex is one of the two vertices the edge connects. An incidence is a pair ( u , e ) {\displaystyle (u,e)} where u {\displaystyle u} is a vertex and e {\displaystyle e} is an edge incident with u {\displaystyle u} .
The edge is said to join and and to be incident on and on . 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.
An incidence in a graph is a vertex-edge pair such that the vertex is an endpoint of the edge. incidence matrix The incidence matrix of a graph is a matrix whose rows are indexed by vertices of the graph, and whose columns are indexed by edges, with a one in the cell for row i and column j when vertex i and edge j are incident, and a zero ...
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 ...
The edge is said to join x and y and to be incident on x and on y. A vertex may exist in a graph and not belong to an edge. The edge (y, x) is called the inverted edge of (x, y). Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head.
An edge-graceful labeling on a simple graph without loops or multiple edges on p vertices and q edges is a labeling of the edges by distinct integers in {1, …, q} such that the labeling on the vertices induced by labeling a vertex with the sum of the incident edges taken modulo p assigns all values from 0 to p − 1 to the vertices.
The edge is removed and its two incident vertices, and , are merged into a new vertex , where the edges incident to each correspond to an edge incident to either or . More generally, the operation may be performed on a set of edges by contracting each edge (in any order).
Appending an edge and a vertex to P 2 gives P 3, the path with three vertices. Denote the vertices by v 1, v 2, and v 3. Label the two edges in the following way: the edge (v 1, v 2) is labeled 1 and (v 2, v 3) labeled 2. The induced labelings on v 1, v 2, and v 3 are then 1, 0, and 2 respectively. This is an edge-graceful labeling and so P 3 ...