On distance integral and distance Laplacian integral graphs

Let [Formula: see text] be a connected graph on [Formula: see text] vertices and let [Formula: see text] and [Formula: see text] be the distance and the distance Laplacian matrices associated with [Formula: see text]. A graph [Formula: see text] is said to be [Formula: see text]-integral (resp. [Formula: see text]-integral) if all eigenvalues of [Formula: see text] (resp. [Formula: see text]) are integers. In this paper, we obtain various conditions under which the graphs [Formula: see text] and [Formula: see text] are distance integral. We also obtain conditions on [Formula: see text], [Formula: see text] under which the dumbbell graph [Formula: see text] is [Formula: see text]-integral.