The characterization of graphs with two trivial distance ideals

TL;DR

This work fully characterizes graphs with at most two trivial distance ideals over $\mathbb{Z}[X]$ and over $\mathbb{Q}[X]$, linking the notion to forbidden induced subgraphs $\{\mathcal{F},\textsf{odd-holes}_7\}$ and providing explicit structural families. It leverages connections between the Smith normal form of distance matrices and distance ideals, proving that connected bipartite graphs have a nontrivial third distance ideal and that third invariant factors are even, hinting at extensions of classical distance-matrix formulas to wider graph classes. The results yield precise classifications for the univariate and rational cases and reveal that blow-ups of base graphs (e.g., complete multipartite constructions) preserve the bounded-trivial-distance-ideal property. Overall, the paper delivers a concrete, implementable taxonomy of graphs with at most two trivial distance ideals and highlights implications for determinant formulas and invariant-factor structure in distance matrices.

Abstract

The distance ideals of graphs are algebraic invariants that generalize the Smith normal form (SNF) and the spectrum of several distance matrices associated with a graph. In general, distance ideals are not monotone under taking induced subgraphs. However, in [7] the characterizations of connected graphs with one trivial distance ideal over $\mathbb{Z}[X]$ and over $\mathbb{Q}[X]$ were obtained in terms of induced subgraphs, where $X$ is a set of variables indexed by the vertices. Later, in [3], the first attempt was made to characterize the family of connected graphs with at most two trivial distance ideals over $\mathbb{Z}[X]$. There, it was proven that these graphs are $\{ \mathcal {F},\textsf{odd-holes}_{7}\}$-free, where $\textsf{odd-holes}_{7}$ consists of the odd cycles of length at least seven and $\mathcal{F}$ is a set of sixteen graphs. Here, we give a characterization of the $\{\mathcal{F},\textsf{odd-holes}_{7}\}$-free graphs and prove that the $\{\mathcal{F},\textsf{odd-holes}_{7}\}$-free graphs are precisely the graphs with at most two trivial distance ideals over $\mathbb{Z}[X]$. As byproduct, we also find that the determinant of the distance matrix of a connected bipartite graph is even, this suggests that it is possible to extend, to connected bipartite graphs, the Graham-Pollak-Lovász celebrated formula $\det(D(T_{n+1}))=(-1)^nn2^{n-1}$, and the Hou-Woo result stating that $\text{SNF}(D(T_{n+1}))=\textsf{I}_2\oplus 2\textsf{I}_{n-2}\oplus (2n)$, for any tree $T_{n+1}$ with $n+1$ vertices. Finally, we also give the characterizations of graphs with at most two trivial distance ideals over $\mathbb{Q}[X]$, and the graphs with at most two trivial distance univariate ideals.

Figures (5)

• Figure 1: The graphs $P_4$, $\sf{paw}$, $\sf{diamond}$ and $C_4$.
• Figure 2: Some minimal forbidden graphs for graphs with 2 trivial distance ideals over $\mathbb{Z}[X]$.
• Figure 3: An induced paw graph in $G$
• Figure 4: Forbidden structures of the 3-leaf power graph.
• Figure 5: Forbidden induce subgraphs for $\Lambda_{2}^{\mathbb{Q}}$

Theorems & Definitions (63)

• Lemma 1
• Lemma 2
• Lemma 3
• Lemma 4
• Proposition 5
• Lemma 6
• Corollary 7
• Theorem 8
• Theorem 9
• Lemma 10