On the Expressive Power of Linear Algebra on Graphs

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 M A T L A N G , 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 M A T L A N G . That is, we are interested in understanding when two graphs cannot be distinguished by posing queries in M A T L A N G on their adjacency matrices. Surprisingly, a complete picture can be painted of the impact of each of the linear algebra operations supported in M A T L A N G 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 show that M A T L A N G -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 M A T L A N G fragment is shown to correspond to equivalence by means of sentences in the two-variable fragment of this logic.