RUS  ENG
Full version
JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2019 Volume 16, Pages 206–216 (Mi semr1050)

This article is cited in 4 papers

Discrete mathematics and mathematical cybernetics

Distance-regular graph with intersection array $\{105,72,24;1,12,70\}$ does not exist

I. N. Belousov, A. A. Makhnev

N.N. Krasovskii Institute of Mathematics and Mechanics of UB RAS, 16, S.Kovalevskaya str., Yekaterinburg, 620990, Russia

Abstract: Distance-regular graph $\Gamma$ of diameter 3 is called Shilla graph if $\Gamma$ containes the second eigenvalue $\theta_1=a_3$. In this case $a=a_3$ devides $k$ and we set $b=b(\Gamma)=k/a$. Koolen and Park obtained the list of intersection arrays for Shilla graphs with $b=3$. A. Brouwer with coauthors proved that graph with intersection array $\{27,20,10;1,2,18\}$ does not exist. $Q$-polinomial Shilla graph with $b=3$ has intersection array $\{42,30,12;1,6,28\}$ or $\{105,72,24;1,12,70\}$. Early authors proved that graph with intersection array $\{42,30,12;1,6,28\}$ does not exist.
We prove that graph with intersection array $\{105,72,24;1,12,70\}$ does not exist.

Keywords: distance-regular graph, Shilla graph, triple intersection numbers.

UDC: 519.17

MSC: 05C25

Received December 18, 2018, published February 8, 2019

DOI: 10.33048/semi.2019.16.012



Bibliographic databases:


© Steklov Math. Inst. of RAS, 2025