We prove new results on perfect state transfer of quantum walks on quotient graphs. Since a graph G has perfect state transfer if and only if its quotient G/π, under any equitable partition π, has perfect state transfer, we exhibit graphs with perfect state transfer between two vertices but which lack automorphism swapping them. This answers a question of Godsil (Discrete Mathematics 312(1):129...

The (n, k)-arrangement graph A(n, k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k − 1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n, k). This equitable partition can be employed to compute the complete set of eigen...

The most widely applied strategy for workload sharing is to equalize the workload assigned to each resource. In mobile multi-agent systems, this principle directly leads to equitable partitioning policies in which (i) the workspace is divided into subregions of equal measure, (ii) there is a bijective correspondence between agents and subregions, and (iii) each agent is responsible for service ...

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particul...

We generalize the stable graph regularity lemma of Malliaris and Shelah to the case of finite structures in finite relational languages, e.g., finite hypergraphs. We show that under the model-theoretic assumption of stability, such a structure has an equitable regularity partition of size polynomial in the reciprocal of the desired accuracy, and such that for each k-ary relation and k-tuple of ...

We prove that a minimal automaton has a minimal adjacency matrix rank and a minimal adjacency matrix nullity using equitable partition (from graph spectra theory) and Nerode partition (from automata theory). This result naturally introduces the notion of matrix rank into a regular language L, the minimal adjacency matrix rank of a deterministic automaton that recognises L. We then define and fo...