A graph G is said to be M-integral (resp. A-integral, D-integral, \(D^L\) -integral or \(D^Q\) -integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G) , distance matrix D(G) , distance Laplacian matrix \(D^L(G)\) or distance signless Laplacian matrix \(D^Q(G)\) ) are integers. Lu et al. [Discrete Math, 346 (2023)] defined the generalized wheel graph GW(a, m, n) as the graph \(aK_{m}\nabla C_{n}\) , and obtained all D-integral generalized wheel graphs \(aK_{m} \nabla C_{n}\) . Based on the above research, in this paper, we determine all \(D^L\) -integral and \(D^Q\) -integral generalized wheel graphs \(aK_{m} \nabla C_{n} \) respectively. As byproducts, we give a sufficient and necessary condition for the join \( G_{1} \nabla G_{2} \) of two regular graphs \( G_{1}\) and \(G_{2} \) to be \( D^{L} \) -integral, from which we can get infinitely many new classes of \( D^{L} \) -integral graphs.
