AI 中文总结
本文建立带极性二部图克罗内克积与极性商的结构兼容性，应用该理论以广义多边形为因子构造出三类大图，其中直径3的低度数图大小超现有同类图。
AI 中文摘要
本文建立了允许极性的二部图的克罗内克积与其极性商之间的结构兼容性，并给出了这类图直径的一个紧上界。对于某些因子图，克罗内克积的直径达到该直径上界，其中包括广义多边形。过去，带极性商的广义多边形已被显著用于构造极大图。我们将本文的结构定理应用于作为因子图的广义多边形$\boldsymbol{G}_n(q,q)$，构建了三个新的大图族，覆盖了无穷但稀疏的度数集合，其中一个直径为2，一个直径为3，一个直径为5。随着广义多边形因子的阶数$q$和$r$增大，这些图渐近接近图大小的理论上界。作为示例，我们开发了一个源自广义四边形的此类族，并构造了新的低度数直径-3图，其大小在对应度数下超过了所有已知图。
英文摘要
In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotient, and provide a sharp upper bound on the diameter of these graphs. For certain factor graphs, the diameter of the Kronecker product meets the upper bound on diameter, among them the generalized polygons. Generalized polygons with their polarity quotients have been notably used in the past to construct very large graphs. We apply the structural theorems in the paper to generalized polygons $\mathbb{G}_n(q,q)$ used as factor graphs, and build three new families of graphs of large order covering an infinite but sparse set of degrees, one of diameter $2$, one of diameter $3$ and one of diameter $5$. These asymptotically approach a theoretical upper bound on graph size as orders $q$ and $r$ of the generalized polygon factors increase. As an example, we develop one such family, derived from generalized quadrangles, and construct new diameter-$3$ graphs of low degree that are larger than any previously known at their degrees.
Comments24 pages, 4 figures, 2 tables
