Distance-regular graphs with intersection arrays $\{104,70,25;1,7,80\}$ and $\{272,210,49;1,15,224\}$ do not exist

M. P. Golubyatnikov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 98-105

Résumé

I. N. Belousov, A. A. Makhnev, and M. S. Nirova in 2019 described $Q$-polynomial distance-regular graphs $\Gamma$ of diameter 3 with strongly regular graphs $\Gamma_2$ and $\Gamma_3$, where the graphs $\Gamma_2$ and $\Gamma_3$ have the same vertices as $\Gamma$ and these vertices are adjacent if and only if they are at distance 2 and 3 in $\Gamma$, respectively. Some of the $Q$-polynomial distance-regular graphs $\Gamma$ with strongly regular graphs $\Gamma_2$ and $\Gamma_3$ have intersection arrays $$\left\lbrace \frac{(s^2+su-1)(u^2-1)}{s^2-1},\frac{(u^2-s^2)su}{s^2-1},u^2;1,\frac{u^2-s^2}{s^2-1},\frac{su^3-su}{s^2-1}\right\rbrace.$$ For small values of $s$ and $u$, we have intersection arrays $\{104,70,25;1,7,80\}$ ($u=5$, $s=2$) and $\{272,210,49;1,15,224\}$ ($u=7$, $s=2$). We prove that distance-regular graphs with such arrays do not exist. We also study the properties of a local subgraph in a hypothetical distance-regular graph with intersection array $\{399, 320, 64; 1, 20, 336\}$ ($u=8$, $s=2$).

Détail

Keywords: distance-regular graph
Mots-clés : $Q$-polynomial graph.