Inverse problems in distance-regular graphs theory
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 $5$.