On distance-regular graphs in which the neighborhood of each vertex is isomorphic to the Hoffman-Singleton graph

A study was conducted to analyze distance-regular graphs in which the neighborhood of each vertex was isomorphic to the Hoffman-Singleton graph. The degree of a vertex was the number of vertices in its neighborhood and the τ was called a regular graph of degree k when the degree of any vertex a in τ was k. τ was also called an edge regular graph with parameters (?, k, λ) when τ was a regular graph of degree k on ? vertices and each of its edges lied in λ triangles. The description for distance regular graphs was completed in which the neighborhoods hoods of vertices are isomorphic to the Hoffman-Singleton graph in the second equation. Automorphisms of the distance regular graphs from the claim of Theorem 1 were investigated in the third equation and it was proved that these graphs were not vertex-transitive.