The enumeration of odd spanning trees in graphs

TL;DR

This paper introduces a unified polynomial-based method to count odd spanning trees in various classes of graphs, providing explicit formulas for several well-known graph families.

Contribution

It develops a novel unified technique using multivariable polynomials to enumerate odd spanning trees across different graph types.

Findings

Derived explicit formulas for odd spanning trees in complete graphs

Extended enumeration methods to complete multipartite and Ferrers graphs

Established a polynomial framework applicable to multiple graph classes

Abstract

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.