The enumeration of odd spanning trees in graphs

A graph is odd if all of its vertices have odd degrees. In particular, an odd spanning tree in a connected graph is a spanning tree in which all vertices have odd degrees. In this paper we establish a unified technique to enumerate odd spanning trees of a graph $G$ in terms of a multivariable polynomial associated with $G$ and indeterminates ${x_{i}:v_i\in V(G)}$. As applications, the enumerative formulas for odd spanning trees in complete graphs, complete multipartite graphs, almost complete graphs, complete split graphs and Ferrers graphs are, respectively, derived from our work.

💡 Research Summary

The paper addresses the problem of counting odd spanning trees—spanning trees in which every vertex has odd degree—in finite, simple, undirected graphs. The authors introduce a unified method based on a multivariate polynomial associated with a graph, denoted (P_G(x_1,\dots,x_n)), where each vertex (v_i) carries an indeterminate (x_i) and each edge ((v_i,v_j)) is weighted by the product (x_i x_j). By the weighted Matrix‑Tree theorem, the cofactor of the weighted Laplacian of (G) equals this polynomial, and expanding it yields \