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DOI: 10.23671/VNC.2016.3.5875 Extensions of Pseudogeometric Graphs for \(pG_{s5}(s,t)\)
Abstract:
J. Koolen posed the problem of studying distanceregular 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 distanceregular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue \(t\) for \(t = 1,2,\ldots\) In the article by A. K. Gutnova and A. A. Makhnev "Extensions of pseudogeometrical graphs for \(pG_{s4} (s, t)\)" 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 distanceregular graphs whose local subgraphs are exceptional pseudogeometric graphs for \(pG_{s5}(s,t)\).
Keywords: distanceregular graph, pseudogeometric graph, eigenvalue of graph
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