We consider the problem of determining the Q–integral graphs, i.e. the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all (2, s)–semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results conce...

Let G be a simple graph and L = L(G) the Laplacian matrix of G. G is called L-integral if all its Laplacian eigenvalues are integer numbers. It is known that every cograph, a graph free of P4, is L-integral. The class of P4-sparse graphs and the class of P4-extendible graphs contain the cographs. It seems natural to investigate if the graphs in these classes are still L-integral. In this paper ...

The existence of perfect state transfer in quantum spin networks based on integral circulant graphs has been considered recently by Saxena, Severini and Shparlinski. Motivated by the mentioned work, Bašić, Petković and Stevanović give the simple condition for the characterization of integral circulant graphs allowing the perfect state transfer in terms of its eigenvalues. They stated that integ...

It is shown that distance powers of an integral Cayley graph over an abelian group Γ are again integral Cayley graphs over Γ. Moreover, it is proved that distance matrices of integral Cayley graphs over abelian groups have integral spectrum.

A graph G is called integral or Laplacian integral if all the eigenvalues of the adjacency matrix A(G) or the Laplacian matrix Lap(G) = D(G)−A(G) of G are integers, where D(G) denote the diagonal matrix of the vertex degrees of G. Let Kn,n+1 ≡ Kn+1,n and K1,p[(p−1)Kp] denote the (n+1)-regular graph with 4n+2 vertices and the p-regular graph with p2 + 1 vertices, respectively. In this paper, we ...

A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs S1(t) = K1,t, S2(n, t), S3(m,n, t), S4(m,n, p, q), S5(m,n), S6(m,n, t), S8(m,n), S9(m,n, p, q), S10(n), S13(m,n), S17(m,n, p, q), S18(n, p, q, t), S19(m,n, p, t), S20(n, p, q) and S21(m, t) are defined. We construct the fifteen classes of larger graphs from the known 15 smaller in...