The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and (f, g) ∈ E if the Hamming distance between f and g is equal to 2 ^k. It is shown that the maximum degree of the graph is equal to 2 ^k(2 ¹+ 1) (2 ²+ 1) ⋯ (2 ^k+ 1) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than 2 ^{2 k + 1}- 2 ^k. It is proven that the graph is connected for 2 k= 2 , 4 , 6 , disconnected for 2 k≥ 10 and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.