A.A. Makhnev, D.V. Paduchikh. A.A. Makhnev, D.V. Paduchikh. Inverse problems in the theory of distance-regular graphs

For a distance-regular graph $\Gamma$ of diameter 3, the graph $\Gamma_i$ can be strongly regular for $i=2$ or $3$. Finding the parameters of $\Gamma_i$ given the intersection array of $\Gamma$ is a direct problem, and finding the intersection array of $\Gamma$ given the parameters of $\Gamma_i$ is the inverse problem. The direct and inverse problems were solved earlier by A.A. Makhnev and M.S. Nirova for $i=3$. In the present paper, we solve the inverse problem for $i=2$: given the parameters of a strongly regular graph $\Gamma_2$, we find the intersection array of a distance-regular graph $\Gamma$ of diameter 3. It is proved that $\Gamma_2$ is not a graph in the half case. We also refine Nirova's results on distance-regular graphs $\Gamma$ of diameter 3 for which $\Gamma_2$ and $\Gamma_3$ are strongly regular. New infinite series of admissible intersection arrays are found: $\{r^2+3r+1,r(r+1),r+2;1,r+1,r(r+2)\}$ for odd $r$ divisible by 3 and
$\{2r^2+5r+2,r(2r+2),2r+3;1,2r+2,r(2r+3)\}$ for $r$ indivisible by $3$ and not congruent to $\pm 1$ modulo $5$.

Keywords: strongly regular graph, distance-regular graph, intersection array.

The paper was received by the Editorial Office on May 11, 2018.

Funding Agency: This work was supported by the Russian Science Foundation (project no. 14-11-00061-П).

Aleksandr Alekseevich Makhnev, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: makhnev@imm.uran.ru.

Dmitrii Viktorovich Paduchikh, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia,
e-mail: dpaduchikh@gmail.com.


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