The biclique partition number of a graph \(G\), denoted \( \operatorname{bp}(G)\), is the minimum number of biclique subgraphs that partition the edge set of \(G\). The Graham-Pollak theorem states that the complete graph on \( n \) vertices cannot be partitioned into fewer than \( n-1 \) bicliques. In this note, we show that for any split graph \( G \), the biclique partition number satisfies \( \operatorname{bp}(G) = \operatorname{mc}(G^c) - 1 \), where \( \operatorname{mc}(G^c) \) denotes the number of maximal cliques in the complement of \( G \). This extends the celebrated Graham-Pollak theorem to a broader class of graphs.

翻译：图 \(G\) 的双团划分数（记为 \( \operatorname{bp}(G)\)）是将 \(G\) 的边集划分为双团子图所需的最少子图数量。Graham-Pollak 定理指出，\( n \) 个顶点的完全图不能划分为少于 \( n-1 \) 个双团。本文证明，对任意分裂图 \( G \)，其双团划分数满足 \( \operatorname{bp}(G) = \operatorname{mc}(G^c) - 1 \)，其中 \( \operatorname{mc}(G^c) \) 表示 \( G \) 的补图中的最大团数量。这一结果将著名的 Graham-Pollak 定理推广至更广泛的图类。