Master circuit matrix

The setting up of a master circuit matrix, whereby all possible circuits of any given graph or network with the same, or a smaller, number of nodes can be obtained as a simple modification of the former, is discussed. With the help of such a master matrix, it is also possible to obtain accurately the number of all possible circuits in the given graph, without actually generating them. Besides the flexibility in the choice of circuits it provides for network studies, the enumeration of all circuits should find useful application in other fields, such as operational research, materials management etc. The method actually exploits the inherent symmetry in a complete graph and is also suited for computer adaptation. In the process, a new method of enumerating all possible circuits of a complete graph, and a formula to determine their number, are given. Finally, some important properties of the master circuit matrix are discussed, which include a formula for the number of circuits in which a given combination of edges occurs, a recurrence relationship for the circuit enumeration of different masters, an expression for the number of circuits in a subgraph of a complete graph and a method for setting up a circuit matrix of an arbitrary graph from master B etc.