Necessary and sufficient conditions for R-graph synthesis

✍ Scribed by David P. Brown; Myril B. Reed

• Publisher: Elsevier Science
• Year: 1962
• Tongue: English
• Weight: 448 KB
• Volume: 273
• Category: Article
• ISSN: 0016-0032
• DOI: 10.1016/0016-0032(62)90711-1

✦ Synopsis

It is shown that any symmetric matrix can be considered as the short circuit admittance matrix of a graph consisting of the union of a complete graph and arbitrary tree form.

The necessary and sufficient conditions that a real matrix of order v-1 be the short circuit admittance matrix of a v-vertex R-graph corresponding to any tree of oriented elements are given. A procedure to determine the corresponding R-graph and tree of oriented elements is also presented. These results are based on a class of matrices which transforms the short circuit admittance matrix of any graph G and tree to the short circuit admittance matrix of G and any other tree. The necessary and sufficient conditions associated with a particular tree form are extended, by way of the transformation matrices, to those for any tree form. A tree in the form of a star is used as a basis of the discussion because of the simplicity of the resulting properties.

An example giving the necessary and sufficient conditions for synthesis of a fourth order matrix as a five vertex R-graph is included to illustrate the results.