Graph Theory, put simply, is the study of lines and points.

It is the sub-field of mathematics which deals with graphs: diagrams that involve points and lines and which often pictorially represent mathematical truths. Graph theory is the study of the relationship between **edges** and **vertices**. Formally, a graph is a pair (V,E), where V is a finite set of vertices and E a finite set of edges.

## Definitions in Graph Theory

Graph Theory has some unique vocabulary:

- An
**arc**is a directed line (a pair of ordered vertices). - An
**edge**is line joining a pair of nodes.**Incident**edges are edges which share a vertex. A edge and vertex are**incident**if the edge connects the vertex to another.

- A
**loop**is an edge or arc that joins a vertex to itself. - A
**vertex**, sometimes called a**node**, is a point or circle. It is the fundamental unit from which graphs are made.**Adjacent**vertices are vertices which are connected by an edge.- The
**degree**of a vertex is simply the number of edges that connect to that vertex. Loops count twice. - A
**predecessor**is the node (vertex) before a given vertex on a path. - A
**successor**is the node (vertex) following a given vertex on a path.

**A walk**is a series of vertices and edges.- A
**circuit**is a closed walk with every edge distinct. - A
**closed walk**is a walk from a vertex back to itself; a series of vertices and edges which begins and ends at the same place. - A
**cycle**is a closed walk with no repeated vertices (except that the first and last vertices are the same). - A
**path**is a walk where no repeated vertices. A**u-v**path is a path beginning at u and ending at v. - A
**u-v walk**would be a walk beginning at u and ending at v.

- A

## More Definitions:

- An
**edge contraction**involves removing an edge from a graph by merging the two vertices it used to join. - In computer science and computer-based graph theory, a
**graph traversal**is an exploration of a graph in which the vertices are visited or updated one by one.

## Types of Graphs

- An
**acyclic directed graph**is a finite directed graph which has no directed cycles. - A
**directed graph**is a graph where the edges have direction; that is, they are ordered pairs of vertices. - A
**condensation**of a multigraph is the graph that results when you delete any multiple edges, leaving just one edge between any two points. - If a graph has a path between every pair of vertices (there is no vertex not connected with an edge), the graph is called a
**connected graph**. - If a graph G’ can be constructed from a graph G by repeated edge contractions or deletions, the graph G’ is a
**graph minor**of G. - An
**inverted graph**G’ of G is a graph with the same vertices but none of the same edges; two vertices in G’ are adjacent if and only if they were not adjacent in G. - A
**multigraph**is a graph without loops, but which may have multiple edges. - A
**null graph**is a graph with no edges. It may have one or more vertices. - An
**oriented graph**is a directed graph that doesn’t have any symmetric pairs of directed edges. - A
**simple graph**is a graph that doesn’t have any loops or multiple edges. No multiple edges means that no two edges have the same endpoints. - A
**subgraph**is a graph whose vertices and edges are included in the vertices and edges of another graph (the**supergraph**). - A
**symmetric graph**is a directed graph D where, for every arc (x,y), the inverted arc (y,x) is also in D. - A
**trivial graph**is a graph with only one vertex. - An
**undirected graph**is a graph where none of the edges have direction; the pairs of vertices that make up each edge are unordered.

## Graph Theory in History

Graph Theory dates back to 1735 and Euler’s Seven Bridges of Königsberg. The city of Königsberg was a town with two islands, connected to each other and to the mainland by seven bridges. The question set was whether it were possible to take a walk and cross each bridge exactly once. In a first demonstration of graph theory, Euler showed that it was not possible.

## Mantel’s Theorem

Mantel’s theorem, published in 1907, tells us the largest number of edges a graph with a given number of vertices may have without having a triangle for a subgraph. It can be stated as:

*A graph with n vertices and m edges will contain a triangle as a subgraph if and only if m > n ^{2}/4*.

## Sources

Introduction to Combinatorics and Graph Theory. Retrieved 8/11/2017 from: https://www.whitman.edu/mathematics/cgt_online/cgt.pdf

Turan’s Theorem

Mathematics of Bioinformatics

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