Topology (electrical Circuits) - Graph Theory - Equivalence

Equivalence

Graphs are equivalent if one can be transformed into the other by deformation. Deformation can include the operations of translation, rotation and reflection; bending and stretching the branches; and crossing or knotting the branches. Two graphs which are equivalent through deformation are said to be congruent.

In the field of electrical networks, there are two additional transforms that are considered to result in equivalent graphs which do not produce congruent graphs. The first of these is the interchange of series connected branches. This is the dual of interchange of parallel connected branches which can be achieved by deformation without the need for a special rule. The second is concerned with graphs divided into two or more separate parts, that is, a graph with two sets of nodes which have no branches incident to a node in each set. Two such separate parts are considered an equivalent graph to one where the parts are joined by combining a node from each into a single node. Likewise, a graph that can be split into two separate parts by splitting a node in two is also considered equivalent.

Read more about this topic:  Topology (electrical Circuits), Graph Theory