There is a long tradition in understanding graphs by investigating their adjacency matrices by means of linear algebra. Similarly, logic-based graph query languages are commonly used to explore graph properties. In this paper, we bridge these two approaches by regarding linear algebra as a graph query language. More specifically, we consider MATLANG, a matrix query language recently introduced, in which some basic linear algebra functionality is supported. We investigate the problem of characterising the equivalence of graphs, represented by their adjacency matrices, for various fragments of MATLANG. That is, we are interested in understanding when two graphs cannot be distinguished by posing queries in MATLANG on their adjacency matrices. Surprisingly, a complete picture can be painted of the impact of each of the linear algebra operations supported in MATLANG on their ability to distinguish graphs. Interestingly, these characterisations can often be phrased in terms of spectral and combinatorial properties of graphs. Furthermore, we also establish links to logical equivalence of graphs. In particular, we 1show that MATLANG-equivalence of graphs corresponds to equivalence by means of sentences in the three-variable fragment of first-order logic with counting. Equivalence with regards to a smaller MATLANG fragment is shown to correspond to equivalence by means of sentences in the two-variable fragment of this logic.

通过线性代数研究图形的邻接矩阵来理解图形有很长的传统。同样，基于逻辑的图查询语言通常用于探索图属性。在本文中，我们通过将线性代数视为图查询语言来桥接这两种方法。更具体地说，我们考虑最近引入的矩阵查询语言M A T L A N G，其中支持了一些基本的线性代数功能。我们研究表征图的等效性的问题，这些图由它们的邻接矩阵表示，用于M A T L A N G的各个片段。也就是说，我们有兴趣了解何时无法通过在相邻矩阵上的M A T L A N G中提出查询来区分两个图。令人惊讶的是，可以完整地描绘出M A T L A N G支持的每个线性代数运算对它们区分图形的能力的影响。有趣的是，这些特征通常可以用谱图的光谱和组合特性来表述。此外，我们还建立了与图的逻辑等价关系的链接。特别是，我们1显示了M A T L A N G图的等价性相当于通过计数的一阶逻辑三变量片段中的句子等价性。关于较小的M A T L A N G片段的等效性通过该逻辑的二元变量片段中的句子显示为等效。