Extensions of Pseudogeometric Graphs for \(pG_{s-5}(s,t)\)

Gutnova A. K. , Makhnev A. A.
Vladikavkaz Mathematical Journal 2016. Vol. 18. Issue 3.

Abstract:
J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue \(\leq t\) for a given positive integer \(t\). This problem is reduced to the description of distance-regular graphs in which neighborhoods of vertices are strongly regular
graphs with non-principal eigenvalue \(t\) for \(t = 1,2,\ldots\) In the article by A. K. Gutnova and A. A. Makhnev "Extensions of pseudogeometrical graphs for \(pG_{s-4} (s, t)\)" the Koolen problem was solved for \(t = 4\) and for pseudogeometrical neighborhoods of vertices. In the article of A. A. Makhnev "Strongly regular graphs with nonprincipal eigenvalue 5 and its extensions" the Koolen problem for \(t = 5\) was reduced to the case where the neighborhoods of vertices are exceptional graphs. In this paper intersection arrays for distance-regular graphs whose local subgraphs are exceptional pseudogeometric graphs for \(pG_{s-5}(s,t)\).

Keywords: distance-regular graph, pseudogeometric graph, eigenvalue of graph

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