Geodesic
On extensions of strongly regular graphs with eigenvalue 4
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 233-255

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J. Koolen posed the problem of studying distance regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue ${}\le t$ for a given positive integer $t$. This problem was solved earlier for $t=3$. A program of studying distance regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue $r$, $3 r\le 4$, was started by the first author in his preceding paper. In this paper, a reduction to local exceptional graphs is performed. In the present work we find parameters of exceptional strongly regular graphs with nonprincipal eigenvalue 4. In addition, we prove that a distance regular graph in which neighborhoods of vertices are exceptional nonpseudogeometric strongly regular graphs with nonprincipal eigenvalue 4 has degree at most 729.
Keywords: graph spectrum, strongly regular graph, distance regular graph. 
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A. A. Makhnev; D. V. Paduchikh. On extensions of strongly regular graphs with eigenvalue 4. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 3, pp. 233-255. http://geodesic.mathdoc.fr/item/TIMM_2015_21_3_a24/