On distance-regular graphs of diameter $3$ with eigenvalue~$0$

A. A. Makhnev ; I. N. Belousov

Matematičeskie trudy, Tome 25 (2022) no. 2, pp. 162-173

Résumé

Graph $\Gamma_i$ for a distance-regular graph $\Gamma$ of diameter $3$ can be strongly regular for $i=2$ or $i=3$. J. Koolen with coauthors found parameters of $\Gamma_2$ by the intersection array of graph $\Gamma$ (independently parameters were obtained by Makhnev A.A. and Paduchikh D.V.). In this case $\Gamma$ has eigenvalue $\theta=a_2-c_3$. In this paper it is consider graphs with eigenvalues $\theta_2=0$ and $\theta_3=a_2-c_3$. It is proved that $\Gamma$ has intersection array $\{yx+yz,yz-y,xy-x;1,x+z,yz\}$. Further if $a_2-c_3\ge -10$ then $\Gamma$ has intersection array $\{12,6,2;1,4,9\}$, $\{60,45,8;1,12,50\}$, $\{63,42,12;1,9,49\}$ or $\{72,45,16; 1,8,54\}$.