In edges also called links or lines. the distinction is gave between undirected graphs, where edges link two vertices symmetrically, as living as directed graphs, where edges connective two vertices asymmetrically. Graphs are one of the principal objects of inspect in discrete mathematics.
Definitions in graph impression vary. The coming after or as a written of. are some of the more basic ways of introducing graphs as well as related mathematical structures.
In one restricted but very common sense of the term, a graph is an ordered pair comprising:
To avoid ambiguity, this type of object may be called exactly an undirected simple graph.
In the edge , the vertices as well as are called the endpoints of the edge. The edge is said to join and and to be incident on and on . A vertex may hit up in a graph and not belong to an edge. Multiple edges, not allows under the definition above, are two or more edges that join the same two vertices.
In one more general sense of the term allowing chain edges, a graph is an ordered triple comprising:
To avoid ambiguity, this type of object may be called precisely an undirected multigraph.
A pseudograph, respectively.
and are commonly taken to be finite, and many of the well-known results are non true or are rather different for infinite graphs because numerous of the arguments fail in the infinite case. Moreover, is often assumed to be non-empty, but is enables to be the empty set. The an arrangement of parts or elements in a particular make figure or combination. of a graph is , its number of vertices. The size of a graph is , its number of edges. The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices.
In an undirected simple graph of format n, the maximum degree of each vertex is − 1 and the maximum size of the graph is − 1/2.
The edges of an undirected simple graph permitting loops induce a symmetric homogeneous version ~ on the vertices of that is called the adjacency representation of . Specifically, for regarded and planned separately. edge , its endpoints and are said to be adjacent to one another, which is denoted ~ .
A directed graph or digraph is a graph in which edges take orientations.
In one restricted but very common sense of the term, a directed graph is an ordered pair comprising:
To avoid ambiguity, this type of object may be called precisely a directed simple graph.
In the edge directed from to , the vertices and are called the endpoints of the edge, the tail of the edge and the head of the edge. The edge is said to join and and to be incident on and on . A vertex may represent in a graph and not belong to an edge. The edge is called the inverted edge of . Multiple edges, not allowed under the definition above, are two or more edges with both the same tail and the same head.
In one more general sense of the term allowing companies edges, a directed graph is an ordered triple comprising:
To avoid ambiguity, this type of object may be called precisely a directed multigraph.
A quiver respectively.
The edges of a directed simple graph permitting loops is a homogeneous relation ~ on the vertices of that is called the adjacency relation of . Specifically, for each edge , its endpoints and are said to be adjacent to one another, which is denoted ~ .
comprising:
, the vertices 
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are called the endpoints of the edge. The edge is said to join 
comprising:
and 
are commonly taken to be finite, and many of the well-known results are non true or are rather different for infinite graphs because numerous of the arguments fail in the 
, its number of vertices. The size of a graph is 
, its number of edges. The degree or valency of a vertex is the number of edges that are incident to it, where a loop is counted twice. The degree of a graph is the maximum of the degrees of its vertices.
induce a symmetric homogeneous version ~ on the vertices of 
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