Abstract:
A Shilla graph is a distance-regular graph $\Gamma$ (with valency $k$) of diameter $3$ that has second eigenvalue $\theta_1$ equal to $a=a_3$. In this case $a$ divides $k$ and the parameter $b=b(\Gamma)=k/a$ is defined. A Shilla graph has intersection array $\{ab,(a+1)(b-1),b_2;1,c_2,a(b-1)\}$. J. Koolen and J. Park showed that for fixed $b$ there are finitely many Shilla graphs. Admissible intersection arrays of Shilla graphs were found for $b\in \{2,3\}$ by Koolen and Park in 2010 and for $b\in \{4,5\}$ by A. A. Makhnev and I. N. Belousov in 2021. Makhnev and Belousov also proved the nonexistence of $Q$-polynomial Shilla graphs with $b=5$ and found $Q$-polynomial Shilla graphs with $b=6$. A $Q$‑polynomial Shilla graph with $b=6$ has intersection array $\{42t,5(7t+1),3(t+3);1,3(t+3),35t\}$ with $t\in \{7,12,17,27,57\}$, $\{372,315,75;1,15,310\}$, $\{744,625,125;1,25,620\}$, $\{930,780,150;1,30,775\}$, $\{312,265,48;$$1,24,260\}$, $\{624,525,80;1,40,520\}$, $\{1794,1500,200;1,100,1495\}$, or $\{5694,4750,600;1,300,4745\}$. The nonexistence of graphs with intersection arrays $\{372,315,75;1,15,310\}$, $\{744,625,125;1,25,620\}$, $\{1794,1500,200;1,$$100,1495\}$, and $\{42t,5(7t+1),3(t+3);1,3(t+3),35t\}$ was proved earlier. We prove that distance-regular graphs with intersection arrays $\{312,265,48;1,24,260\}$, $\{624,525,80;1,40,520\}$, and $\{930,780,150;1,30,775\}$ do not exist.
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