Recent trends and future directions in vertex-transitive graphs

A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade

Cross products, invariants, and centralizers

An algebra V with a cross product x has dimension 3 or 7. In this work, we use 3-tangles to describe, and provide a basis for, the space of homomorphisms from V-circle times n to V-circle times m that are invariant under the action of the automorphism group Aut(V, x) of V, which is a special orthogonal group when dim V = 3, and a simple algebraic group of type G(2) when dim V = 7. When m = n, this gives a graphical description of the centralizer algebra End(Aut(v, x))(V-circle times n), and therefore, also a graphical realization of the Aut(V, x)-invariants in V-circle times 2n equivalent to the First Fundamental Theorem of Invariant Theory. We show how the 3-dimensional simple Kaplansky Jordan superalgebra can be interpreted as a cross product (super)algebra and use 3-tangles to obtain a graphical description of the centralizers and invariants of the Kaplansky superalgebra relative to the action of the special orthosymplectic group