Homomorphism Tensors and Linear Equations

TL;DR

Grohe, Rattan, and Seppelt develop a unified algebraic framework tying homomorphism indistinguishability to linear equation feasibility via tensor maps. They show that for graph classes such as pathwidth, treedepth, and treewidth, indistinguishability is captured by systems PW$^{k+1}$, TD$^k$, or related relaxations, linking these to logical fragments and Sherali–Adams hierarchies. Their approach leverages a duality between tensor maps and graph-equivalence, underpinned by Specht–Wiegmann-type results for involution monoids and bilabelled-graph calculus, yielding new characterisations and a partial negative result for bounded-degree trees. The results unify existing theorems and provide new tools to analyze graph classes, cyclewidth, pathwidth, treewidth, and treedepth, with potential extensions to metric variants and broader graph families.

Abstract

Lovász (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over several natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018), and graphs of bounded treedepth.

Figures (6)

• Figure 1: Example for a tree $t \in \Gamma_M$ for $M = [5]$. The root is depicted in grey. Here, $t_{\mathsf A} = (A_2^*((A_1\boldsymbol{1}) \odot (A_5\boldsymbol{1}))) \odot (A_3 \boldsymbol{1})$.
• Figure 2: The (bi)labelled graphs from \ref{['ex:adjacency']} in wire notation of mancinska_quantum_2020: A vertex carries in-label (out-label) $i$ if it is connected to the number $i$ on the left (right) by a wire. Actual edges and vertices of the graph are depicted in black.
• Figure 3: Combinatorial operations on bilabelled graphs.
• Figure 4: Interplay of a family of labelled graphs $\mathcal{R} \subseteq \mathcal{G}(k)$ and a family of bilabelled graphs $\mathcal{S} \subseteq \mathcal{G}(k,k)$ yielding matrix equations with orthogonal, pseudo-stochastic, or doubly stochastic solutions. Arrows indicate implications, e.g. every gluing-closed family of labelled graphs $\mathcal{R}$ is inner-product compatible.
• Figure 5: Bilabelled graphs in $\mathcal{B}^k$ as defined in \ref{['lem:basal']}.

Theorems & Definitions (128)

• Theorem 1.1
• Theorem 1.1
• Theorem 1.1
• Lemma 2.1
• proof
• Lemma 2.3: Vandermonde Interpolation
• proof
• Lemma 2.4: tinhofer_graph_1986
• proof
• Theorem 2.5: Frobenius--Schur frobenius_uber_1906