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.