Generation and realisation of loop and cutsets

A relationship between the transformations of trees and the corresponding basic loop and cutset matrices is established, which facilitates generation of these matrices with respect to any other or all possible trees of a graph in a routine manner. These transformation matrices can be set up directly without resorting to a series of elementary transformations. This method of generating the basic loop and cutset matrices is based on the concept of a Lagrangian tree. A method for minimising the number of nonzero entries in a loopset matrix is also discussed. Finally, two new methods for realising a given matrix as a loop or cutset matrix are proposed. These methods of analysis and synthesis will be particularly useful in dealing with complex networks which are commonly encountered in communication systems, electrical and switching networks, and models of many other physical systems.