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Shilla distance-regular graphs with $b_2 = sc_2$
I. N. Belousovab a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
A Shilla graph is a distance-regular graph
$\Gamma$ of diameter 3 whose second eigenvalue is
$a=a_3$. A Shilla graph has intersection array
$\{ab,(a+1)(b-1),b_2;1,c_2,a(b-1)\}$. J. Koolen and J. Park showed that, for a given number
$b$, there exist only finitely many Shilla graphs. They also found all possible admissible intersection arrays of Shilla graphs for
$b\in \{2,3\}$. Earlier the author together with A.A. Makhnev studied Shilla graphs with
$b_2=c_2$. In the present paper, Shilla graphs with
$b_2=sc_2$, where
$s$ is an integer greater than
$1$, are studied. For Shilla graphs satisfying this condition and such that their second nonprincipal eigenvalue is
$-1$, five infinite series of admissible intersection arrays are found. It is shown that, in the case of Shilla graphs without triangles in which
$b_2=sc_2$ and
$b<170$, only six admissible intersection arrays are possible. For a
$Q$-polynomial Shilla graph with
$b_2=sc_2$, admissible intersection arrays are found in the cases
$b=4$ and
$b=5$, and this result is used to obtain a list of admissible intersection arrays of Shilla graphs for
$b\in\{4,5\}$ in the general case.
Keywords:
distance-regular graph, graph automorphism.
UDC:
519.17
MSC: 05C25 Received: 20.02.2018
DOI:
10.21538/0134-4889-2018-24-3-16-26
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