The enumeration of odd spanning trees in graphs

TL;DR

This paper develops a unified framework to enumerate odd spanning trees in graphs by introducing a vertex-weighted multivariable spanning-tree polynomial $P_G(x_1,\dots,x_n)$ and linking it to the weighted Laplacian via the Matrix-Tree theorem. The central result expresses the number of odd spanning trees $\tau_o(G)$ of a connected graph with even order as $\tau_o(G)=\frac{1}{2^{n}}\sum_{\boldsymbol{\sigma}\in\{\pm1\}^{n}}(\prod_i \sigma_i) P_G(\boldsymbol{\sigma})$, enabling a single framework to count such trees across diverse graph classes. The authors apply this technique to complete graphs, complete multipartite graphs, almost complete graphs, complete split graphs, and Ferrers graphs, deriving explicit closed-form formulas in each case. This work provides a powerful, general method for parity-restricted spanning-tree enumeration and suggests avenues for extending the approach to HIST generalizations and other degree-restriction problems.

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.

Figures (2)

• Figure : Fig. 1. The Ferrers diagram (left) and Ferrers graph (right) corresponding to the partition $(5,5,3,2,1)$.
• Figure : Fig. 2. All non-isomorphic unlabeled odd trees of order 10: $T_i$ for $i\in \{1,2,\ldots,7\}$.

Theorems & Definitions (27)

• Theorem 1.1: Z1
• Theorem 1.2: F1
• Theorem 2.1: B1L1K3
• Theorem 2.2: H3
• Remark 2.3
• Theorem 2.4
• proof
• Theorem 3.1: K1
• proof : Proof of Theorem \ref{['x1-2']}
• Theorem 3.2: K1