Weighted variants of the Andr\'asfai-Erd\H{o}s-S\'os Theorem

A well known result due to Andr'asfai, Erd\H{o}s, and S'os asserts that for $r\ge 2$ every $K_{r+1}$-free graph on $n$ vertices with $\delta (G)>\frac{3r-4}{3r-1}n$ is $r$-partite. We study related questions in the context of weighted graphs, which are motivated by recent work on the Ramsey-Tur'an problem for cliques.