D-Integral, \(D^Q\)-Integral and \(D^L\)-Integral Generalized Double-Wheel Graphs

The graph \(aK_{m,m}\nabla C_{n}\) $a{K}_{m,m}\mathrm{\nabla }{C}_n$ is named the generalized double-wheel graph. A graph G is said to be M-integral (resp. A-integral, D-integral, \(D^L\) $D^L$ -integral or \(D^Q\) $D^Q$ -integral) if all eigenvalues of its matrix M (resp. adjacency matrix A(G) , distance matrix D(G) , distance Laplacian matrix \(D^L(G)\) $D^L(G)$ or distance signless Laplacian matrix \(D^Q(G)\) $D^Q(G)$ ) are integers. In this paper, we completely determine all D-integral, \(D^L\) $D^L$ -integral and \(D^Q\) $D^Q$ -integral generalized double-wheel graphs respectively.