Geodesic
Three infinite families of Shilla graphs do not exist
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 45-50.

Voir la notice de l'article provenant de la source Math-Net.Ru

A distance-regular graph of diameter 3 with the second eigenvalue $\theta_1=a_3$ is called a Shilla graph. For a Shilla graph $\Gamma$, the number $a=a^3$ divides $k$ and we set $b=b(\Gamma)=k/a$. Three infinite families of Shilla graphs with the following admissible intersection arrays were found earlier: $\{b(b^2-1),b^2(b-1),b^2;1,1,(b^2-1)(b-1)\}$ (I.N. Belousov), $\{b^2(b-1)/2,(b-1)(b^2-b+2)/2,b(b-1)4;1,b(b-1)/4,b(b-1)^2/2\}$ (Koolen, Park), and $\{(s+1)(s^3-1),s^4,s^3;1,s^2,s(s^3-1)\}$. In this paper, it is proved that, in the first family, there exists a unique graph, namely, a generalized hexagon of order 2, whereas there are no graphs in the second or third families.
Keywords: distance-regular graph, Shilla graph, triple intersection numbers. 
@article{DANMA_2021_498_a8,
     author = {A. A. Makhnev and I. N. Belousov and M. P. Golubyatnikov and M. S. Nirova},
     title = {Three infinite families of {Shilla} graphs do not exist},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {45--50},
     publisher = {mathdoc},
     volume = {498},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2021_498_a8/}
}
TY  - JOUR
AU  - A. A. Makhnev
AU  - I. N. Belousov
AU  - M. P. Golubyatnikov
AU  - M. S. Nirova
TI  - Three infinite families of Shilla graphs do not exist
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2021
SP  - 45
EP  - 50
VL  - 498
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DANMA_2021_498_a8/
LA  - ru
ID  - DANMA_2021_498_a8
ER  - 
%0 Journal Article
%A A. A. Makhnev
%A I. N. Belousov
%A M. P. Golubyatnikov
%A M. S. Nirova
%T Three infinite families of Shilla graphs do not exist
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2021
%P 45-50
%V 498
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DANMA_2021_498_a8/
%G ru
%F DANMA_2021_498_a8
A. A. Makhnev; I. N. Belousov; M. P. Golubyatnikov; M. S. Nirova. Three infinite families of Shilla graphs do not exist. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 498 (2021), pp. 45-50. http://geodesic.mathdoc.fr/item/DANMA_2021_498_a8/

[1] Brouwer A.E., Cohen A.M., Neumaier A., Distance-Regular Graphs, Springer-Verlag, B.–Heidelberg–N.Y., 1989, 489 pp. | MR | Zbl

[2] Koolen J.H., Park J., “Shilla distance-regular graphs”, Europ. J. Comb., 31 (2010), 2064–2073 | DOI | MR | Zbl

[3] Makhnev A.A., Nirova M.S., “Distantsionno-regulyarnye grafy Shilla”, Mat. zametki, 103:5 (2018), 730–744 | DOI | MR | Zbl

[4] Belousov I.N., Makhnev A.A., “To the theoty of Shilla graphs with ${{b}_{2}} = {{c}_{2}}$”, Sib. Electron. Math. Reports, 14 (2017), 1135–1146 | MR | Zbl

[5] Belousov I.N., “Distantsionno-regulyarnye grafy Shilla s ${{b}_{2}} = s{{c}_{2}}$”, Trudy IMM UrO RAN, 24, no. 3, 2018, 16–26

[6] Coolsaet K., Jurishich A., “Using equality in the Krein conditions to prove nonexistence of sertain distance-regular graphs”, J. Comb. Theory, Series A, 115 (2008), 1086–1095 | DOI | MR | Zbl

[7] Gavrilyuk A., Koolen J., “A characterization of the graphs of bilinear $d \times d$-forms over ${{F}_{2}}$”, Combinatorica, 39:2 (2019), 289–321 | DOI | MR | Zbl