V.V. Bitkina, A.K. Gutnova. On Shilla graphs with $b=6$ and $b_2\ne c_2$ ... P. 74-83

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 $Q$\nobreakdash-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.

Keywords: Shilla graph, distance-regular graph, $Q$-polynomial graph

Received February 17, 2022

Revised April 28, 2022

Accepted April 30, 2022

Funding Agency: This study supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-02-2022-890).

Viktoriya V. Bitkina, Cand. Phys.-Math. Sci., North Ossetian State University, Vladikavkaz, 362025 Russia, e-mail: bviktoriyav@mail.ru

Alina K. Gutnova, Cand. Phys.-Math. Sci., North Ossetian State University, Vladikavkaz, 362025 Russia, e-mail: gutnovaalina@gmail.com

REFERENCES

1.   Brouwer A.E., Cohen A.M., Neumaier A. Distance-regular graphs. Berlin; Heidelberg; NY: Springer-Verlag, 1989, 495 p. ISBN: 0387506195 .

2.   Koolen J.H., Park J. Shilla distance-regular graphs. Europ. J. Comb., 2010, vol. 31, no. 8, pp. 2064–2073. doi: 10.1016/j.ejc.2010.05.012 

3.   Makhnev A.A., Belousov I.N. Shilla graphs with b = 5 and b = 6. Ural Math. J., 2021, vol. 7, no. 2, pp. 51–58. doi: 10.15826/umj.2021.2.004 

4.   Jurisic A., Vidali J. Extremal 1-codes in distance-regular graphs of diameter 3. Des. Codes Cryptogr., 2012, vol. 65, pp. 29–47.

5.   Gavrilyuk A.L., Koolen J. A characterization of the graphs of bilinear (d × d)-forms over $\mathbb {F}_2$. Combinatorica, 2019, vol. 39, no. 2, pp. 289–321. doi: 10.1007/s00493-017-3573-4 

Cite this article as: V.V. Bitkina, A.K. Gutnova. On Shilla graphs with $b=6$ and $b_2\ne c_2$, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 74–83.