Comparing the numbers of subforests and subgraph-degree-tuples

TL;DR

This paper explores the enumeration of certain matrix row-column sums and their connection to acyclic subgraphs in grid graphs, extending the correspondence to broader graph classes and proposing new conjectures.

Contribution

It establishes a novel link between matrix enumeration and acyclic subgraph counts, and formulates conjectures relating degree sequences to subgraph counts in bipartite graphs.

Findings

Count of specific matrix row-column sums matches acyclic subgraph numbers.

Conjecture relating degree sequences of subgraphs to acyclic subgraph counts.

Proven cases for cactus and generalized book graphs.

Abstract

We enumerate the row-column-sums of all square tridiagonal $(0,1)$-matrices and prove that their count coincides with OEIS A022026 $-$ the number of acyclic subgraphs of the complete $2\times n$ grid graph. We then extend this correspondence in two independent directions: 1. admitting larger sets of matrix entries, and 2. relaxing the tridiagonal support to broader prescribed sparsity patterns. The latter leads us to conjecture that, for any bipartite graph $G$, the number of its acyclic subgraphs equals the number of degree sequences realized by subgraphs of $G$. Moreover, for any non-bipartite graph, the former should be strictly smaller than the latter. We discuss several general approaches and prove these hypotheses for cactus graphs and generalized book graphs.