Graph: a representation of a circuit where each branch is denoted by a line segment.
Tree (of a graph): a set of branches (each denoted by a line segment) that connects every node to every other node via some path without forming a loop.
Tree branch: a branch of a graph that is part of a particular tree.
Cotree: those branches of a graph which are not part of a particular tree. This is also known as the complement of the tree.
Link: a branch of a cotree.
Cut set: a minimum set of branches that, when cut, will divide a graph into two separate parts.
Fundamental cut set: a cut set containing only a single tree branch.
Fundamental loop: A loop that results when a link is put into the tree.
A cut set divides a graph into two independent parts. In terms of the original circuit, a KCL equation can be written for either part of the circuit divided by the cut; such a KCL equation is called a cut-set equation.
The dual of a fundamental cut set is a fundamental loop. Each time a link is inserted into a tree as a potential tree branch, a loop is formed in the tree (thus the resulting object is no longer a tree). Such a loop is called a fundamental loop.
Tree-branch analysis uses KCL (Kirchhoff's Current Law) but no reference node is selected like is done in nodal analysis; all KCL equations are written in terms of tree branch voltages instead of node voltages.
Loop analysis uses KVL (Kirchhoff's Voltage Law) but the loops chosen may not necessarily be meshes. Instead, each loop needs to be a fundamental loopobtained by inserting a link into a tree.
There is only one graph for a circuit (although there may be many ways to draw it).
Usually, there are several trees for a graph, and each tree has a corresponding cotree.