We study perfect state transfer on Cayley graphs from the point of view that state transfer is a property of a graph and not of a group. This paper is a bridge between the classical question about isomorphic Cayley graphs of non-isomorphic groups and quantum walks on graphs. We show that a Cayley graph of a group with an abelian subgroup of index two is a Cayley graph of an abelian group under any one of three hypotheses, two drawn from the theory of isomorphic Cayley graphs. A statement of the same kind holds for extraspecial groups: every Cayley graph of an extraspecial p-group of order p^2n+1 with a conjugacy-closed connection set is a Cayley graph of Z_p^2n+1. From these results we deduce that every explicit construction of perfect state transfer in the six papers we survey, on dihedral, dicyclic, generalized dihedral, V_8n and extraspecial 2-groups, is a non-abelian presentation of an abelian Cayley graph. Moreover, we show that a non-abelian group with an abelian subgroup of index two admits a connected Cayley graph with perfect state transfer if and only if its order is divisible by four. Genuinely non-abelian examples do exist. We prove that, for every odd prime power q≥ 5, the SL(2,q) graph of Pantangi and Sin, which they showed to admit perfect state transfer, is a Cayley graph of no abelian group; to our knowledge, this is the first infinite family of Cayley graphs with perfect state transfer provably admitting no abelian Cayley presentation. We also construct an infinite family of Cayley graphs with peak state transfer and determine all regular subgroups of the automorphism group of every member. An appendix records a census of the connected vertex-transitive graphs with perfect state transfer on at most 30 vertices.

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd eigenvalue.

Lovász et al. proved that every $6$-edge-connected graph has a nowhere-zero $3$-flow. In fact, they proved a more technical statement which says that there exists a nowhere zero $3$-flow that extends the flow prescribed on the incident edges of a single vertex $z$ with bounded degree. We extend this theorem of Lovász et al. to allow $z$ to have arbitrary degree, but with the additional assumption that there is another vertex $x$ with large degree and no small cut separating $x$ and $z$. Using this theorem, we prove two results regarding the generation of minimal graphs with the property that prescribing the edges incident to a vertex with specific flow does not extend to a nowhere-zero $3$-flow. We use this to further strengthen the theorem of Lovász et al., as well as make progress on a conjecture of Li et al.

Evra, Feigon, Maurischat, and Parzanchevski (2023) introduced a biregular extension of Cayley graphs. In this paper, we reformulate their definition and provide some basic properties. We also show how these Cayley incidence graphs relate to various notions of Cayley hypergraphs. We further establish connections between Cayley incidence graphs and certain geometric and combinatorial structures, including coset geometries, difference sets and cages.

Sabidussi's theorem [Duke Math. J. 28 (1961), 573–578] gives necessary and sufficient conditions under which the automorphism group of a lexicographic product of two graphs is a wreath product of the respective automorphism groups. We prove a quantum version of Sabidussi's theorem for finite graphs, with the automorphism groups replaced by quantum automorphism groups and the wreath product replaced by the free wreath product of quantum groups. This extends the result of Chassaniol [J. Algebra 456, 2016, 23–45], who proved it for regular graphs. Moreover, we apply our result to lexicographic products of quantum vertex transitive graphs, determining their quantum automorphism groups even when Sabidussi's conditions do not apply.

A quantum Latin square is an n × n array of unit vectors where each row and column forms an orthonormal basis of a fixed complex vector space. We introduce the notion of (G,G')-invariant quantum Latin squares for finite groups G and G'. These are quantum Latin squares with rows and columns indexed by G and G' respectively such that the inner product of the a,b-entry with the c,d-entry depends only on a^-1c ∈ G and b^-1d ∈ G'. This definition is motivated by the notion of group invariant bijective correlations introduced in [Roberson & Schmidt (2020)], and every group invariant quantum Latin square produces a group invariant bijective correlation, though the converse does not hold. In this work we investigate these group invariant quantum Latin squares and their corresponding correlations. Our main result is that, up to applying a global isometry to every vector in a (G,G')-invariant quantum Latin square, there is a natural bijection between these objects and trace and conjugate transpose preserving isomorphisms between the group algebras of G and G'. This in particular proves that a (G,G')-invariant quantum Latin square exists if and only if the multisets of degrees of irreducible representations are equal for G and G'. Another motivation for this line of work is that whenever Cayley graphs for groups G and G' are quantum isomorphic, then there is a (G,G')-invariant quantum correlation witnessing this, and thus it suffices to consider such correlations when searching for quantum isomorphic Cayley graphs. Given a group invariant quantum correlation, we show how to construct all pairs of graphs for which it gives a quantum isomorphism.

Sabidussi's theorem [Duke Math. J. 28, 1961] gives necessary and sufficient conditions under which the automorphism group of a lexicographic product of two graphs is a wreath product of the respective automorphism groups. We prove a quantum version of Sabidussi's theorem for finite graphs, with the automorphism groups replaced by quantum automorphism groups and the wreath product replaced by the free wreath product of quantum groups. This extends the result of Chassaniol [J. Algebra 456, 2016], who proved it for regular graphs. Moreover, we apply our result to lexicographic products of quantum vertex transitive graphs, determining their quantum automorphism groups even when Sabidussi's conditions do not apply.

A connected, locally finite graph Γ is a Cayley–Abels graph for a totally disconnected, locally compact group G if G acts vertex-transitively with compact, open vertex stabilizers on Γ. Define the minimal degree of G as the minimal degree of a Cayley–Abels graph of G. We relate the minimal degree in various ways to the modular function, the scale function and the structure of compact open subgroups. As an application, we prove that if T_d denotes the d-regular tree, then the minimal degree of Aut(T_d) is d for all d≥ 2.

In this paper, we characterize perfect state transfer in Cayley graphs for abelian groups that have a cyclic Sylow-2-subgroup. This generalizes a result of Bašić from 2013 where he provides a similar characterization for Cayley graphs of cyclic groups.

We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.

We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible 2-ended graphs. We draw connections to Willis’ scale function and re-prove a result by Trofimov.