Graph and Trees

A graph $G$ is a mathematical object composed of points known as vertices or nodes and lines connecting some (possibly empty) subset of them, known as edges. Formally, a graph is a binary relation on a set of vertices. If this relation is symmetric, the graph is said to be undirected; otherwise, the graph is said to be directed. Graphs in which at most one edge connects any two nodes are said to be simple graphs. Vertices are usually not allowed to be self-connected, but this restriction is sometimes relaxed to allow such "loops". The edges of a graph may be assigned specific values or labels, in which case the graph is called a labeled graph. Unless stated otherwise, the unqualified term "graph" usually refers to a simple graph. A non-simple graph with no loops but which can contain more than one edge between any two points is called a multigraph.