Homomorphism Tensors and Linear Equations

TL;DR

This paper develops an algebraic framework based on tensors to unify and analyze homomorphism indistinguishability across various graph classes, linking it to linear algebra and representation theory.

Contribution

It introduces a unified algebraic approach to characterize homomorphism indistinguishability over multiple graph classes using tensor and linear transformation analysis.

Findings

Characterizes homomorphism indistinguishability for bounded degree trees.

Provides algebraic conditions for graphs of bounded pathwidth.

Answers open question on bounded treedepth graph classes.

Abstract

Lov\'asz (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.