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DOI: 10.46698/y2738-1800-0363-i

# Distance-Regular Graphs with Intersection Arrays {7,6,6;1,1,2} and {42,30,2;1,10,36} Do not Exist

Makhnev, A. A. , Bitkina, V. V. , Gutnova, A. K.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4.
Abstract:
Let $$\Gamma$$ be a distance-regular graph of diameter $$3$$ without triangles, $$u$$ be a vertex of the graph~$$\Gamma$$, $$\Delta^i =\Gamma_i (u)$$ and $$\Sigma^i = \Delta^i_{2,3}$$. Then $$\Sigma^i$$ is a regular graph without $$3$$-cocliques of degree $$k'=k_i-a_i-1$$ on $$v' = k_i$$ vertices. Note that for non-adjacent vertices $$y, z \in \Sigma^i$$ we have $$\Sigma^i = \{y, z\} \cup \Sigma^i (y) \cup \Sigma^i (z)$$. Therefore, for $$\mu'= |\Sigma^i (y) \cap \Sigma^i (z)|$$ we have the equality $$v'= 2k' + 2-\mu'$$. Hence the graph $$\Sigma$$ is coedge regular with parameters $$(v', k', \mu')$$. It is proved in the paper that a distance-regular graph with intersection array $$\{7,6,6; 1,1,2 \}$$ does not exist. In the article by M. S. Nirova "On distance-regular graphs with $$\theta_2 = -1$$" is proved that if there is a strongly regular graph with parameters $$(176,49,12,14)$$, in which the neighborhoods of the vertices are $$7 \times 7$$ -lattices, then there also exists a distance-regular graph with intersection array $$\{7,6,6; 1,1,2\}$$. M. P. Golubyatnikov noticed that for a distance-regular graph $$\Gamma$$ with intersection array $$\{7,6,6; 1,1,2\}$$ graph $$\Gamma_2$$ is distance regular with intersection array $$\{42,30,2; 1,10,36\}$$. With this result and calculations of the triple intersection numbers, it is proved that the distance-regular graphs with intersection arrays $$\{7,6,6; 1,1,2\}$$ and $$\{42,30,2; 1,10,36\}$$ do not exist.
Keywords: distance-regular graph, triangle-free graph, triple intersection numbers