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# Graph Theory

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 Sub Topics Graph theory would be the study of charts, which are mathematical structures utilized to model pairwise relationships between objects. A "graph" in this context consist of "vertices" or "nodes" and also lines called sides that connect these vertices.A graph could be undirected, meaning that there is no distinction between your two vertices associated with each edge, or its edges could be directed from one vertex to an alternative:  Graphs are one of many prime objects involving study under discrete mathematics.Graphs are manifested visually simply by drawing a dot or perhaps circle for any vertex, and pulling an arc concerning two vertices whenever they are connected by a good sharp edge. If the actual graph is actually directed, the actual direction is actually indicated simply by drawing the arrow.

## Definition

A graph is a symbolic representation of an network and of their connectivity. It indicates an abstraction in the reality so it can be simplified as a few linked nodes.
Graph theory is really a branch regarding mathematics interested in how networks can be encoded and their components measured.

A chart G is a few vertex (nodes) v connected by simply edges (links) e. Thus G=(v, e).

## Tree Graph Theory

Within mathematics, and far more specifically in graph theory, a tree is definitely an undirected graph by which any two vertices usually are connected by exactly one simple path.

A tree is definitely an undirected simple graph G that satisfies a few of the following equivalent situation:

1) G is connected and it has no cycles.
2) G doesn't need any cycles, and a simple cycle is built if any edge is joined with G.
3) G is definitely connected, but seriously isn't connected if virtually any single edge is removed from G.
4) G is connected as well as the 3-vertex complete graph K$_3$ seriously isn't a minor of G.
Any two vertices in G might be connected by an original simple path.

When G has finitely quite a few vertices, say n of these, then the above statements may also be equivalent to some of the following conditions:

G is connected and it has n − 1 perimeters.
G has no simple cycles and it has n − 1 perimeters.

## Path Graph Theory

Path graph or linear graph is really a particularly simple example of any tree, namely a tree with a couple of vertices that is not branched at all, that is, includes only vertices connected with degree 2 and 1.

In specific, it has a couple terminal vertices (vertices which have degree 1), while all others (if any) get degree 2.
Paths and cycles tend to be fundamental concepts connected with graph theory,
A path is really a trail in which often all vertices (except probably the first and past ones) are distinct.

A path among two vertices u and v is termed a u-v path.
The list of vertices and edges which head over to make up a path form a sub graph. This sub graph itself is also called a path.
Open Path
An open path is a path in which the first and last vertices are distinctive.
If the very first and last vertices are classified as the same, a path is known as a cycle.