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.

翻译：Lovász（1967）证明了两个图$G$和$H$同构当且仅当它们在所有图类上无法通过同态计数区分，即对于每个图$F$，从$F$到$G$的同态数量等于从$F$到$H$的同态数量。近年来，在受限图类（如有界树宽、有界树深和平面图）上的同态不可区分性，已发展为一个颇具影响力的统一框架，用于刻画由逻辑等价性和代数方程组所引发的多种图等价关系。本文通过分析标记图同态计数张量的线性代数与表示论结构，为上述结果提供了一个统一的代数框架。同态张量子空间之间存在特定线性变换，该存在性既可解释为图类上的同态不可区分性，也可视为方程系统的可解性。基于这一框架，我们获得了若干自然图类（即有界度树、有界路径宽图（回答了Dell等人（2018）提出的问题）以及有界树深图）上同态不可区分性的刻画。