Let \(\varGamma \) be a distance-regular graph of diameter 3 with strong regular graph \(\varGamma _3\). The determination of the parameters \(\varGamma _3\) over the intersection array of the graph \(\varGamma \) is a direct problem. Finding an intersection array of the graph \(\varGamma \) with respect to the parameters \(\varGamma _3\) is an inverse problem. Previously, inverse problems were solved for \(\varGamma _3\) by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph \(\varGamma \) of diameter 3, for which the graph \({\bar{\varGamma }}_3\) is a pseudo-geometric graph of the net \(PG_{m}(n, m)\). New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array \(\{20,16,5; 1,1,16 \}\).

中文翻译：

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图论中的反问题：网络

令\（\ varGamma \）为直径3的距离正则图，且具有强正则图\（\ varGamma _3 \）。在图\（\ varGamma \）的交集数组上确定参数\（\ varGamma _3 \）是一个直接的问题。相对于参数\（\ varGamma _3 \）找到图\（\ varGamma \）的交集是一个反问题。以前，Makhnev和Nirova为\（\ varGamma _3 \）解决了反问题。在本文中，我们研究了直径为3的距离规则图形\（\ varGamma \）的交集，其中图形\（{\ bar {\ varGamma}} _ 3 \）是网络\（PG_ {m}（n，m）\）的伪几何图。找到了这些图的新的无限系列的可允许相交阵列。我们还研究了具有交集\（\ {20,16,5; 1,1,16 \} \）的距离正则图的自同构。