The number of spanning trees in an arbitrary graph or multigraph is obtained via a general formula involving eigenvalues of an associated matrix, This is shown to be particularly useful in the case of graphs (or multigraphs) which are joins, and a method for deriving the appropriate eigenvaiues of joins is given. As applications of this, concise general expressions are derived for the number of spanning trees in any wheel or top. For the ordinary n spoke wheel W_{n+1} = K_{1} + C_{n} the simple formula {\Pi|_{r=1}^{n-1} (3-2 \cos (2r{\pi}/n)) is derived. A more general concept of join of multigraphs is introduced, and this is applied to obtain a simple formula in terms of integers and cosines for the number of spanning trees in general multigraph wheels.

General solution to the spanning tree enumeration problem in arbitrary multigraph joins

Comment on "General Solution to the Spanning Tree Enumeration Problem in Multigraph Wheels

Enumerating functions which, compared with existing techniques, greatly facilitate counting the spanning trees of certain classes of graphs are presented, with examples of their application. One of these functions gives the spanning tree count in terms of the spanning forests of the subgraph which results on separating, from the original ordinary graph G , the complete bipartite subgraph defined by the incidence set of any one of its vertices. Another is the function derived by removing two disjoint incidence sets of a certain type from the graph.

Spanning tree enumeration by bipartite subgraph separation

A network-theoretic approach for counting the number of spanning trees of a graph is proposed. This approach is based on a theorem in the theory of determinants. Following this approach, a recurrence relation for counting Γn, the number of spanning trees in a multigraph ladder having (n+1) nodes, is established. A recurrence relation is obtained connecting the sequences {Wn} and {Γn} where Wn is the number of spanning trees in a multigraph wheel having (n+1) nodes. The significance of the approach is further illustrated by giving simple proofs of certain well-known results, in particular, the formula for counting the number of spanning trees in a cascade of 2-port networks.

http://ieeexplore.ieee.org/iel7/6449389/6451876/06591843.pdf

A theorem in the theory of determinants and the number of spanning trees in a graph

https://ieeexplore.ieee.org/iel5/31/23453/01084176.pdf

Comments on 'General expressions for network complexity of selected families of graphs'

A recurrence relation and an explicit solution for the number of spanning trees in a multigraph wheel with unequal numbers of spokes and rim edges are given. Previous results can be identified as special cases of the general enumeration and the solution for one new specific case is included.

http://ieeexplore.ieee.org/iel5/8147/23425/01083621.pdf

N. Bose

Papers

General solution to the spanning tree enumeration problem in multigraph wheels