Abstract
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 \(\{20,16,5; 1,1,16 \}\).
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This work is supported by RSF, project 14-11-00061-\({\Pi }\), Research of the third author was supported by the NNSF of China (11771409).
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Makhnev, A.A., Golubyatnikov, M.P. & Guo, W. Inverse Problems in Graph Theory: Nets. Commun. Math. Stat. 7, 69–83 (2019). https://doi.org/10.1007/s40304-018-0159-4
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DOI: https://doi.org/10.1007/s40304-018-0159-4