Georgia Institute of Technology College of Sciences

Fox et al introduced the model of c-closed graphs, a distribution-free model motivated by one of the most universal signatures of social networks, triadic closure. Even though enumerating maximal cliques in an arbitrary network can run in exponential time, it is known that for c-closed graph, enumerating maximal cliques and maximal complete bipartite graphs is always fast, i.e., with complexity being polynomial in the number of vertices in the network. In this work, we investigate further by enumerating maximal blow-ups of an arbitrary graph H in c-closed graphs. We prove that given any finite graph H, the number of maximal blow-ups of H in any c-closed graph on n vertices is always at most polynomial in n. When considering maximal induced blow-ups of a finite graph H, we provide a characterization of graphs H when the bound is always polynomial in n. A similar general theorem is also proved when H is infinite.

We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role  in  systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified  model for an energy harvesting device, in which context the energy growth translates into generation of electricity.
Joint work with S.W. Akingbade and T-M. Seara.

Recent research trends have explored curious analogies between the theory of graphs and Riemann surfaces. To each graph we can associate a real metric torus, known as its Jacobian. It was previously known that isomorphisms of graph Jacobians yield isomorphisms of the associated graphic matroids, partially mirroring a classical algebraic geometry result known as the Torelli theorem. However, the result on graphs is not as strong as a direct analogue of the Riemann surface result would be, nor does it use as much data. I will discuss how the graph Torelli theorem can be refined to incorporate additional data as with Riemann surfaces, in which case it produces isomorphisms of graphs. If time permits, I will describe further recent work in this direction.

Let H be a graph. We show that if r is large enough as a function of H,
then the r-partite Turán graph maximizes the number of copies of H among
all Kr+1-free graphs on a given number of vertices. This confirms a
conjecture of Gerbner and Palmer.