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. We then obtain a recurrence relation 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.