In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotients, and provide a sharp upper bound on the diameter of these graphs. We use these properties to construct new families of graphs, all of which have orders that asymptotically approach the Moore bound. We show that the bipartite compatibility graphs have a natural polarity, and that the corresponding polarity quotients comprise a new family of graphs of any even diameter $2k$ with degree $(q+1)^k$ for any prime power $q$, much smaller than those of other known families of that diameter and field order $q$. These families generalize the Erdős-Rényi-Brown polarity graphs. Kronecker products of these polarity quotients give another new family of graphs of any even diameter for every pair of prime powers $q,r$. Certain generalized polygons are known to admit polarity with polarity quotients approaching the Moore bound asymptotically. Kronecker products of these polarity quotients give another new family of graphs of diameters $2$, $3$ and $5$, including a diameter-$3$ degree-$18$ graph of order larger than any previously known for that degree and diameter.

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