DISTANCE-REGULAR GRAPH WITH INTERSECTION ARRAY {27, 20, 7; 1, 4, 21} DOES NOT EXIST

Konstantin S. Efimov     (Ural State University of Economics, 62 March 8th Str., Ekaterinburg, 620144; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russian Federation)
Alexander A. Makhnev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russia, Russian Federation)

Abstract


In the class of distance-regular graphs of diameter 3 there are 5 intersection arrays of graphs with at most 28 vertices and noninteger eigenvalue. These arrays are \(\{18,14,5;1,2,14\}\), \(\{18,15,9;1,1,10\}\), \(\{21,16,10;1,2,12\}\), \(\{24,21,3;1,3,18\}\), and \(\{27,20,7;1,4,21\}\). Automorphisms of graphs with intersection arrays \(\{18,15,9;1,1,10\}\) and \(\{24,21,3;1,3,18\}\) were found earlier by A.A. Makhnev and D.V. Paduchikh. In this paper, it is proved that a graph with the intersection array \(\{27,20,7;1,4,21\}\) does not exist.


Keywords


Distance-regular graph, Graph \(\Gamma\) with strongly regular graph \(\Gamma_3\), Automorphism

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References


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DOI: http://dx.doi.org/10.15826/umj.2020.2.006

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