# Graph

The net is vast and infinite

Basic

• A Graph (G) is made up of a set V of vertices and a set E of edges.
• Another names for the vertices are nodes and another name for the edges is links.
• The order of the graph (|V|) is the number of vertices.
• The size of the graph (|E|) is the number of edges.
• A vertex is listed as letter on this page. Ex a graph contain the vertices a, b and c.
• A edge is listed as as the two vertices it connect. Ex (a,b) and (b,c).
• In a undirected graph the edge (a,b) = (b,a).
• A directed graph have edges that go in one direction from a source node to a destination node.

Walks

• A walk is a sequence of vertices. A walk is closed if the first vertex and the last are the same, and open if they are different.
• A open walk with no repeat vertices is known as a path.
• A closed walk with no repeat vertices is known as a cycle.
• A acyclic graph contains no cycles.
• The length of a walk is the number of edges.

Tree

• A acyclic graph is called a tree.
• A spanning tree of a graph is another graph that contains all the vertices and is a tree. If each edge has a length then the tree with the minimal sum of all the weights is called the minimum spanning tree.

• The degree of a vertex is the number of edges it has. Loops are counted twice.
• A vertex of degree 0 is a isolated vertex and one with degree 1 is a leaf.
• The total degree of a graph is the sum of degrees of all it's vertices.
• Two vertices a and b are called adjacent if they are connected by a edge.

Other

• A graph is said to be a weighted graph if it's edges contains a value.
• A complete graph is a graph where there is a edge between every pair of vertices.
• Both the nodes and the edges can contain information.
• A graph can be said to be between sparse or dense. Sparse graph have few connections per node and dense graphs have many.