To the theory of Shilla graphs with $b_2=c_2$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1135-1146
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In this paper by using exact formulas for multiplicities of eigenvalues it is
founded new infinite serie intersection arrays of $Q$-polynomial Shilla graph with
$b_2 = c_2$. Intersection array of $Q$-polynomial Shilla graph $\Gamma$ with $b_2=c_2$ is
$\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}$ and for any vertex $u\in \Gamma$
the subgraph $\Gamma_3(u)$ is an antipodal distance-regular graph with the intersection array
$\{t(2r+1),(2r-1)(t+1),1;1,t+1,t(2r+1)\}$.
In case $t=2r^2-1$ the intersection array is feasible and in case $t=r(2lr-(l+1))$
the intersection array is feasible only if $(l,r)\in \{(1,2),(2,1),(4,1),(6,1)\}$.
Keywords:
distance-regular graph, Shilla graph.
@article{SEMR_2017_14_a78,
author = {A. A. Makhnev and I. N. Belousov},
title = {To the theory of {Shilla} graphs with $b_2=c_2$},
journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
pages = {1135--1146},
publisher = {mathdoc},
volume = {14},
year = {2017},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/SEMR_2017_14_a78/}
}
A. A. Makhnev; I. N. Belousov. To the theory of Shilla graphs with $b_2=c_2$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 14 (2017), pp. 1135-1146. http://geodesic.mathdoc.fr/item/SEMR_2017_14_a78/