Connectivity of 3-distance graphs

For a simple graph [Formula: see text], the [Formula: see text]-distance graph, [Formula: see text], is a graph with the vertex set [Formula: see text] and two vertices are adjacent if and only if the distance between them is [Formula: see text] in the graph [Formula: see text]. For a connected graph [Formula: see text], we provide some conditions for the connectedness of [Formula: see text]. Also, we characterize all trees and unicyclic graphs with connected [Formula: see text]-distance graphs.