Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 65-69
Voir la notice de l'article provenant de la source Math-Net.Ru
Distance-regular graph $\Gamma$ of diameter $3$ having the second eigenvalue $\theta_1=a_3$ is called Shilla graph. In this case $a=a_3$ devides $k$ and we set $b=b(\Gamma)=k/a$. Jurishich and Vidali found intersection arrays of $Q$-polynomial Shilla graphs with $b_2=c_2$: $\{2rt(2r+1),(2r-1)(2rt+t+1),r(r+t);1,r(r+t),t(4r^2-1)\}$. But many arrays in this series are not feasible. Belousov I. N. and Makhnev A. A. found a new infinite series feasible arrays of $Q$-polynomial Shilla graphs with $b_2=c_2$ ($t=2r^2-1$): $\{2r(2r^2-1)(2r+1),(2r-1)(2r(2r^2-1)+2r^2),r(2r^2+r-1);1,r(2r^2+r-1),(2r^2-1)(4r^2-1)\}$. If $r=2$ then we have intersection array $\{140,108,18;1,18,105\}$. In the paper it is proved that graph with this intersection array does not exist.
@article{VMJ_2021_23_2_a4,
author = {A. A. Makhnev and M. S. Nirova},
title = {Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist},
journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
pages = {65--69},
publisher = {mathdoc},
volume = {23},
number = {2},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VMJ_2021_23_2_a4/}
}
TY - JOUR
AU - A. A. Makhnev
AU - M. S. Nirova
TI - Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist
JO - Vladikavkazskij matematičeskij žurnal
PY - 2021
SP - 65
EP - 69
VL - 23
IS - 2
PB - mathdoc
UR - http://geodesic.mathdoc.fr/item/VMJ_2021_23_2_a4/
LA - ru
ID - VMJ_2021_23_2_a4
ER -
%0 Journal Article
%A A. A. Makhnev
%A M. S. Nirova
%T Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist
%J Vladikavkazskij matematičeskij žurnal
%D 2021
%P 65-69
%V 23
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VMJ_2021_23_2_a4/
%G ru
%F VMJ_2021_23_2_a4
A. A. Makhnev; M. S. Nirova. Distance-regular graph with intersection array $\{140,108,18;1,18,105\}$ does not exist. Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 2, pp. 65-69. http://geodesic.mathdoc.fr/item/VMJ_2021_23_2_a4/