Adjacency-degree algebras and spectral determination of graphs

Abstract

Let $G$ be a graph with adjacency matrix $A$ and degree matrix $D$. Applying $A$ and $D$ repeatedly to the all-ones vector produces a space $M_G$ of functions on the vertices, and these functions are constant on the orbits of the automorphism group. We call $G$ AD-observable if $M_G$ contains the indicator vector of every orbit, and AD-separable if the functions in $M_G$ separate the orbits. Forests are AD-observable, by McKay's leaf-pruning argument. Our main result is that for every cograph, $M_G$ is spanned by the indicator vectors of the cells of the stable partition of color refinement. Hence a cograph is AD-observable, or equivalently AD-separable, exactly when color refinement finds its orbits, and we describe these cographs by their cotrees. We also show that two cographs with the same number of vertices and the same homomorphism counts from all caterpillars have the same counts from all trees. The main result does not extend to distance-hereditary graphs: we give two such graphs on nine vertices that are AD-separable but not AD-observable.

arXiv commentThe main results are reformulated from v1 and results on cographs are added