I.N. Belousov, A.A. Makhnev. Inverse problems in the class of Q-polynomial graphs ... P. 14-22

In the class of distance-regular graphs $\Gamma$ of diameter 3 with a pseudogeometric graph $\Gamma_3$, feasible intersection arrays for the partial geometry were found for networks by Makhnev, Golubyatnikov, and Guo; for dual networks by Belousov and Makhnev; and for generalized quadrangles by Makhnev and Nirova. These authors obtained four infinite series of feasible intersection arrays of distance-regular graphs:
$$\big\{c_2(u^2-m^2)+2c_2m-c_2-1,c_2(u^2-m^2),\ (c_2-1)(u^2-m^2)+2c_2m-c_2;1,c_2,u^2-m^2\big\},$$ $$\{mt,(t+1)(m-1),t+1;1,1,(m-1)t\}\ \ \text{for}\ \  m\le t,$$ $$\{lt,(t-1)(l-1),t+1;1,t-1,(l-1)t\},\ \ \text{and}\ \   \{a(p+1),ap,a+1;1,a,ap\}.$$ We find all feasible intersection arrays of $Q$-polynomial graphs from these series. In particular, we show that, among these infinite families of feasible arrays, only two arrays ($\{7,6,5;1,2,3\}$ (folded 7-cube) and $\{191,156,153;1,4,39\}$) correspond to $Q$-polynomial graphs.

Keywords: distance-regular graph, $Q$-polynomial graph, graph $\Gamma$ with a strongly regular graph $\Gamma_3$

Received May 22, 2020

Revised June 17, 2020

Accepted July 13, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research – the National Natural Science Foundation of China (project no. 20-51-53013_a).

Ivan Nikolaevich Belousov, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia,; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: i_belousov@mail.ru

Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci., RAS Corresponding Member, Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, e-mail: makhnev@imm.uran.ru

REFERENCES

1.   Bang S., Koolen J. Distance-regular graphs of diameter three having eigenvalue -1. Linear Algebra and its Applications, 2017, vol. 531, pp. 38–53. doi: 10.1016/j.laa.2017.05.038 

2.   Iqbal Q., Koolen J., Park J., Rehman M. Distance-regular graphs with diameter 3 and eigenvalue $a_2-c_3$. Linear Algebra and Appl., 2020, vol. 587, pp. 271–290. doi: 10.1016/j.laa.2019.10.021 

3.   Makhnev A.A., Golubyatnikov M.P., Guo Wenbin. Inverse problems in distance-regular graphs: nets. Communications in Mathematics and Statistics, 2019, vol. 7, no. 1, pp. 69–83. doi: 10.1007/S40304-018-0159-4 

4.   Makhnev A.A., Nirova M.S. Inverse problems in distance-regular graphs: generalized quadrangles. Sibirean Electr. Math. Reports, 2018, vol. 15, pp. 927–934. doi: 10.17377/semi.2018.15.079 

5.   Brouwer A.E., Cohen A.M., Neumaier A. Distance-Regular Graphs. Berlin; Heidelberg; N Y: Springer-Verlag, 1989, 495 p. ISBN: 0387506195 .

6.   Terwilliger P. A new inequality for distance-regular graphs. Discrete Mathematics, 1995, vol. 137, pp. 319–332. doi: 10.1016/0012-365X(93)E0170-9 

7.   Jurisic A., Vidali J. Extremal 1-codes in distance-regular graphs of diameter 3. Des. Codes Cryptogr., 2012, vol. 65, no. 1-2, pp. 29–47. doi: 10.1007/s10623-012-9651-0 

Cite this article as: I.N. Belousov, A.A. Makhnev. Inverse problems in the class of Q-polynomial graphs, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 14–22.