We use a novel approach to construct n-deltahedra (n ? 14). Solving two Diophantine equations, we obtain vertex sets of deltahedra that may be either planar or non-planar graphs. By using recursive processes we construct planar graphs of deltahedra. Also, by using K5 or K3,3 we build non-planar graphs. We then construct Laplacian matrix of order m and obtain the spectra, 0 = λ1 ? λ2 ? … ? λm of n-deltahedra, n = 4, 6, 8, 10, 12, and 14–deltahedra. We found the interesting properties of λm = m, and of the second smallest eigenvalue λ2. We also show that the details of the complement of graph, its eigenvalues, λ1 = 0, and the eigenvalues of the complement of graph, λi = m – λm-i+2, the eigenvalues of original graph.
