The Distance Spectral Radius of the Complements of Graphs with Given Connectivity

Let [Formula: see text] be a simple connected graph with vertex set [Formula: see text]. The distance between two vertices [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the length of a shortest path connecting them in [Formula: see text]. The distance matrix of [Formula: see text], denoted by [Formula: see text], is the [Formula: see text] matrix [Formula: see text]. Then the distance matrix of a simple connected graph [Formula: see text] is a symmetric matrix with real eigenvalues. The maximum eigenvalue is called the distance spectral radius of [Formula: see text], written as [Formula: see text]. In this paper, we characterize the unique graph whose complement attains the maximum distance spectral radius among all graphs with given connectivity. We also establish bounds for the distance spectral radius of such graph complements.