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J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue ≤ t for the given positive integer t. This problem is reduced to the des­cription of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the non-principal eigenvalue t for t = 1, 2,… In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, A.A. Makhnev and D.V. Paduchikh found the arrays of intersections of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue t such as 2 < t ≤ 3. The graphs with intersection arrays {125,96,1;1,48,125}, {176,150,1;1,25,176}, and {256,204,1;1,51,256} remain unexplored. In this paper, we have found the possible orders and the structures of subgraphs of the fixed points of automorphisms of a distance-regular graph with the intersection array {125,96,1;1,48,125}. It has been proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4,6). Composition factors of the automorphism group of a distance-regular graph with the intersection array {125,96,1;1,48,125} have been determined.

The paper deals with the optimal control problem, including the linear parabolic equation as a state problem. Pointwise constraints are imposed on the control function. The objective functional involves the observation function in the entire space-time domain. The optimal control problem is approximated by a finite dimensional problem with mesh approximation of the state equation by the explicit (forward Euler) mesh scheme with variable time steps. The existence of unique solutions for the continuous and mesh optimal control problems is proved. The Uzawa-type iterative method is used for solving the finite dimensional optimal control problem. The results of numerical experiments are presented.