The enumeration of spanning tree of weighted graphs

Abstract

A polynomial associated with G is defined as $$t(G,w)=\sum _{T\in {\mathbb {T}}(G)}\prod _{e\in E(T)}w_e(G)$$ ( $${\mathbb {T}}(G)$$ is the set of spanning trees of G), which is a weighted enumeration of spanning trees of graphs. It is known that any graph G is an intersection graph of a linear hypergraph, which corresponds to a clique partition of G. In this paper, we introduce the Schur complement formula and local transformation formula of t(G, w). By using these formulas, we obtain some expressions of t(G, w) for weighted intersection graphs and express the number of spanning trees of graph G in terms of clique partitions of G. As applications, expressions for enumerating spanning trees in bipartite graphs, line graphs, generalized line graphs, middle graphs, total graphs, generalized join graphs and vertex-weighted graphs are derived from our work.