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 two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).

翻译：Lov\'asz(1967) 显示,两张G$和$H$的图形,如果而且只有在所有图表类中,即每张G$,均态数量从F美元到$G美元,均态数量等于从F美元到$H美元。最近,两张G$和$H美元是非形态化的,如果两张G$和$H$是不可分化的,那么,它们作为在所有图表类中获取不同等同关系的惊人强大框架,这些图表是不可区分的,也就是说,每张图类和平面方形系统产生的图,我们为这些结果提供了一个统一的平面化框架,通过检查从标注的图形中计算同质化的强压体的线性和代表性理论性结构。这种同质化的调和性阵列之间的某些线性转变,可以被解释为对一个图形类的均态化,以及一个公式化的平面系统的可行性,也就是说,在两个平面的平面的平面结构中,我们获得了两个平面性平面的平面的平面结构。