Geodesic
On $Q$-polynomial Shilla graphs with $b=6$
Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 117-123 
A. A. Makhnev; Zhigang Wan. On $Q$-polynomial Shilla graphs with $b=6$. Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 117-123. http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a9/
@article{VMJ_2022_24_2_a9,
     author = {A. A. Makhnev and Zhigang Wan},
     title = {On $Q$-polynomial {Shilla} graphs with $b=6$},
     journal = {Vladikavkazskij matemati\v{c}eskij \v{z}urnal},
     pages = {117--123},
     year = {2022},
     volume = {24},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a9/}
}
TY  - JOUR
AU  - A. A. Makhnev
AU  - Zhigang Wan
TI  - On $Q$-polynomial Shilla graphs with $b=6$
JO  - Vladikavkazskij matematičeskij žurnal
PY  - 2022
SP  - 117
EP  - 123
VL  - 24
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a9/
LA  - ru
ID  - VMJ_2022_24_2_a9
ER  - 
%0 Journal Article
%A A. A. Makhnev
%A Zhigang Wan
%T On $Q$-polynomial Shilla graphs with $b=6$
%J Vladikavkazskij matematičeskij žurnal
%D 2022
%P 117-123
%V 24
%N 2
%U http://geodesic.mathdoc.fr/item/VMJ_2022_24_2_a9/
%G ru
%F VMJ_2022_24_2_a9

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. For such graph $a=a_3$ devides $k$ and we set $b = b(\Gamma) = k/a$. Further $a_1 = a - b$ and $\Gamma$ has intersection array $\{ab,(a + 1)(b - 1), b_2; 1, c_2, a(b - 1)\}$. I. N. Belousov and A. A. Makhnev found feasible arrays of $Q$-polynomial Shilla graphs with $b=6$: $\{42t,5(7t+1),3(t+3);1,3(t+3),35t\}$, where $t\in \{7,12,17,27,57\}$, $\{312,265,48;1,24,260\}$, $\{372,315,75;1,15,310\}$, $\{624,525,80;1,40,520\}$, $\{744,625,125;1,25,620\}$, $\{930,780,150;1,30,775\}$, $\{1794,1500,200;1,100,1495\}$ or $\{5694, 4750,600;1,300,4745\}$. It is proved in the paper that graphs with intersection arrays $\{372,315,75;1,15,310\}$, $\{744,625,125;1,25,620\}$ and $\{1794,1500,200;1,100,1495\}$ do not exist.

[1] Brouwer A. E., Cohen A. M., Neumaier A., Distance-Regular Graphs, Springer-Verlag, Berlin–Heidelberg–New York, 1989 | MR | Zbl

[2] Koolen J. H., Park J., “Shilla Distance-Regular Graphs”, European Journal of Combinatorics, 31:8 (2010), 2064–2073 | DOI | MR | Zbl

[3] Belousov I. N., Makhnev A. A., “Shilla Graphs with $b = 5$ and $b = 6$”, Ural Mathematical Journal, 7:2 (2021), 51–58 | DOI | MR | Zbl

[4] Coolsaet K., Jurishich A., “Using Equality in the Krein Conditions to Prove Nonexistence of Certain Distance-Regular Graphs”, Journal of Combinatorial Theory, Series A, 115:6 (2008), 1086–1095 | DOI | MR | Zbl

[5] Gavrilyuk A. L., Koolen J. H., “A Characterization of the Graphs of Bilinear $(d\times d)$-Forms over $\mathbb{F}_2$”, Combinatorica, 39:2 (2010), 289–321 | DOI | MR