The number of spanning trees in an iterative contact graph composed of general cells

Abstract

AbstractThis paper derives a formula for the number of spanning trees in a contact graph with an iterative structure having k general graphs as cells. Mainly, the cascade, wheel, and prism types of contact graphs which are representative among them, are considered. Until now, G has been restricted to an m‐ cycle Cm or n‐point complete graph Kn, and the calculating formula of electrical networks has been applied to unit‐conductance networks of these contact graphs. Then by considering the equi‐electric potential property, etc., the number of spanning trees with k and m or n being variables has been derived. When cells are general G, however, there are no variables corresponding to m and n because of the instability of its structure. Therefore, in this paper we use the formula between conductance of each branch of a network obtained by equivalence transformation of G into K3 or K4 with respect to vertices of G concerned with contact, that is, contact points and the numbers of spanning trees of G and graphs constructed by shorting contact point pairs of G. Also, we represent traditional point‐contact and branch‐contact in the extensive manner of s‐contact by adding s‐multiple branches to contact portions of contact graphs. In other words, the number of spanning trees of these contact graphs is given by a function of the numbers of spanning trees of G and graphs obtained by shorting specific contact point pairs of G and k and s. Furthermore, this result can be applied directly to the derivation of the formula for the summation of all possible tree admittance products in an admittance network which has no mutual coupling with the forementioned graphs as its interconnection structure.