Inverse Problems in the Theory of Distance-Regular Graphs: Dual 2-Designs

Let  $$\Gamma$$ be a distance-regular graph of diameter 3 with a strongly regular graph  $$\Gamma_{3}$$ . Finding the parameters of  $$\Gamma_{3}$$ from the intersection array of  $$\Gamma$$ is a direct problem, and finding the intersection array of  $$\Gamma$$ from the parameters of  $$\Gamma_{3}$$ is its inverse. The direct and inverse problems were solved by A.A. Makhnev and M.S. Nirova: if a graph  $$\Gamma$$ with intersection array $$\{k,b_{1},b_{2};1,c_{2},c_{3}\}$$ has eigenvalue $$\theta_{2}=-1$$ , then the graph complementary to  $$\Gamma_{3}$$ is pseudo-geometric for $$pG_{c_{3}}(k,b_{1}/c_{2})$$ . Conversely, if  $$\Gamma_{3}$$ is a pseudo-geometric graph for $$pG_{\alpha}(k,t)$$ , then  $$\Gamma$$ has intersection array $$\{k,c_{2}t,k-\alpha+1;1,c_{2},\alpha\}$$ , where $$k-\alpha+1\leq c_{2}t

Keywords:

• Mathematics
• Dual (category theory)
• Inverse problem
• Discrete mathematics

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