# Inverse Problems in Graph Theory: Nets

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## 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 \(\{20,16,5; 1,1,16 \}\).

## Keywords

Distance-regular graph Pseudo-geometric graph Strong regular graph## Mathematics Subject Classification

05C25 05E30## References

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