Nonexistence of certain Q-polynomial distance-regular graphs

A. A. Makhnev ; M. P. Golubyatnikov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 4, pp. 136-141

Résumé

I. N. Belousov, A. A. Makhnev, and M. S. Nirova described $Q$-polynomial distance-regular graphs $\Gamma$ of diameter 3 for which the graphs $\Gamma_2$ and $\Gamma_3$ are strongly regular. Set $a=a_3$. A graph $\Gamma$ has type (I) if $c_2+1$ divides $a$, type (II) if $c_2+1$ divides $a+1$, and type (III) if $c_2+1$ divides neither $a$ nor $a+1$. If $\Gamma$ is a graph of type (II), then $a+1=w(c_2+1)$, $t^2=w(w(c_2+1)+c_2)$, and either (i) $w=s^2$, $t^2=s^2(s^2(c_2+1)+c_2)$, $(s^2(c_2+1)+c_2$ is the square of an integer $u$, $c_2=(u^2-s^2)/(s^2+1)$, $t=su$, and $a=(u^2s^2-1)/(s^2+1)$ or (ii) $c_2=sw$, $t^2=w^2(sw+1+s)$, $sw+1+s$ is the square of an integer $u$, $c_2=(u^2-1)w/(w+1)$, $t=uw$, $a=(u^2w^2-1)/(w+1)$, and $\Gamma$ has intersection array $$\left\{ \frac{u^3w^2+u^2w^2+uw-1}{w+1},\frac{(u^2-1)uw^2}{w+1},\frac{(u^2w+1)w}{w+1};1,\frac{(u^2-1)w}{w+1},\frac{(u^2w+1)uw}{w+1}\right\}.$$ If a graph of type (IIii) is such that $w=u$, then it has intersection array $\{w^4+w-1,w^4-w^3,(w^2-w+1)w;$ $1,w(w-1),(w^2-w+1)w^2\}$. We prove that graphs with such intersection arrays do not exist for even $w$.

Détail

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