Closeness Centralization Measure for Two-mode Data of Prescribed Sizes
  We confirm a conjecture by Everett, Sinclair, and Dankelmann~[Some Centrality results new and old, J. Math. Sociology 28 (2004), 215--227] regarding the problem of maximizing closeness centralization in two-mode data, where the number of data of each type is fixed. Intuitively, our result states that among all networks obtainable via two-mode data, the largest closeness is achieved by simply locally maximizing the closeness of a node. Mathematically, our study concerns bipartite graphs with fixed size bipartitions, and we show that the extremal configuration is a rooted tree of depth~$2$, where neighbors of the root have an equal or almost equal number of children. 
Matja\v{z} Krnc, Jean-S\'ebastien Sereni, Riste \v{S}krekovski,   Zelealem B. Yilma
math.CO