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Graph Theory:

Graph theory, in computer science and mathematics is the study of graphs, which are mathematical structures utilized to model pair-wise relations among objects from a certain collection. In this context a "graph" is a collection of "vertices" or "nodes" and a collection of edges that connect pairs of vertices. A graph might be undirected, meaning that there is no distinction among the two vertices linked with each edge, or its edges might be directed from one vertex to another; see graph (mathematics) for more thorough definitions and for other variations in the types of graph that are considered commonly. Graphs are one of the key objects of study in discrete mathematics.

A graph is an ordered pair G = (V, E) in which:

V is the vertex set whose elements are the vertices, or nodes of the graph. This set is frequently denoted as V (G) or just V.
E is the edge set which elements are the edges, or connections among vertices, of the graph. This set is frequently denoted E (G) or just E. If the graph is undirected, particular edges are unordered pairs {u, v} where u and v are vertices in V. If the graph is directed, edges are ordered pairs (u, v).

Two graphs G and H are assume equal when V (G) = V(H) and E(G) = E(H).

The order of a graph the number of vertices in it, usually indicated by |V| or sometimes n. The size of graph is the number of edges present in it, mentioned by |E| , or sometimes m . If n = 0 or m = 0, the graph is called empty or null. If n = 1 the graph is considered trivial.

Undirected graph:

It is one where edges have no orientation. The edge (a, b) is identical to the edge (b, a),that means as they are not ordered pairs, but sets {u, v} (or 2-multisets) of vertices.

Directed graph:

A directed graph or digraph is an ordered pair D = (V, A) with following

‘V’ a set whose elements are called as vertices or nodes, and
‘A’ a set of ordered pairs of vertices, called as arcs, directed edges, or arrows.

An arc a = (x, y) is assumed to be directed from x to y; y is said the head and x is said the tail of the arc; y is called to be a direct successor of x, and x is refer to be a direct predecessor of y. If any path leads from x to y, then y is called to be a successor of x and accessible from x, and x is said to be a predecessor of y. The arc (y, x) is said the arc (x, y) inverted.

A directed graph D is said symmetric if, for each arc in D, the corresponding inverted arc also connected to D. A symmetric loop-less directed graph D = (V, A) is corresponding to a simple undirected graph G = (V, E), where the pairs of inverse arcs in A correspond 1-to-1 along with the edges in E; thus the edges in G number |E| = |A|/2, or half the number of arcs in D.

On this definition a variation is the oriented graph, in which not more than one of (x, y) and (y, x) can be arcs.

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