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

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 Γ_n, the number of spanning trees in a multigraph ladder having (n+1) nodes, is established. A recurrence relation is obtained connecting the sequences {W_n} and {Γ_n} where W_n 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.