Subgraph densities in signed graphons and the local Sidorenko conjecture

We prove inequalities between the densities of various bipartite subgraphs in signed graphs and graphons. One of the main inequalities is that the density of any bipartite graph with girth r cannot exceed the density of the r-cycle. This study is motivated by Sidorenko's conjecture, which states that the density of a bipartite graph F with m edges in any graph G is at least the m-th power of the edge density of G. Another way of stating this is that the graph G with given edge density minimizing the number of copies of F is, asymptotically, a random graph. We prove that this is true locally, i.e., for graphs G that are "close" to a random graph.Comment: 20 page

On a Theorem of Sewell and Trotter

Sewell and Trotter [J. Combin. Theory Ser. B, 1993] proved that every connected alpha-critical graph that is not isomorphic to K_1, K_2 or an odd cycle contains a totally odd K_4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K_4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter's theorem.Comment: Referee comments incorporate

The automorphism group of a graphon

We study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on $k$-tuples of points. Among applications we study the graph algebras defined by finite rank graphons and the space of node-transitive graphons.Comment: 29 pages, 2 figure

The minimal density of triangles in tripartite graphs

We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomass\'e and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from $K_{3,3,3}$ has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.Comment: 44 pages including 12 page appendix of C++ cod