Suppose that G is a simple, vertex-labeled graph and that S is a multiset. Then if there exists a one-to-one mapping between the elements of S and the vertices of G, such that edges in G exist if and only if the absolute difference of the corresponding vertex labels exist in S, then G is an autograph, and S is a signature for G. While it is known that many common families of graphs are autographs, and that infinitely many graphs are not autographs, a non-autograph has never been exhibited. In this paper, we identify the smallest non-autograph: a graph with 6 vertices and 11 edges. Furthermore, we demonstrate that the infinite family of graphs on n vertices consisting of the complement of two non-intersecting cycles contains only non-autographs for n ≥ 8.
Within sports analytics, there is substantial interest in comprehensive statistics intended to capture overall player performance. In baseball, one such measure is wins above replacement (WAR), which aggregates the contributions of a player in each facet of the game: hitting, pitching, baserunning, and fielding. However, current versions of WAR depend upon proprietary data, ad hoc methodology, and opaque calculations. We propose a competitive aggregate measure, openWAR, that is based on public data, a methodology with greater rigor and transparency, and a principled standard for the nebulous concept of a “replacement” player. Finally, we use simulation-based techniques to provide interval estimates for our openWAR measure that are easily portable to other domains.