The initial question : which graphs are determined by their spectrum originates from chemistry. And the critical problem is the existence of cospectral graphs. This problem is related to the fundamental problem of the graph isomorphism. The point is simple. Since checking whether two graphs are cospectral can be done in polynomial time, we can restrict the graph isomorphism to cospectral graphs. As van Dam and Haemers pointed out, we can use the following equivalences:

1. A and B are cospectral
2. A and B have the same characteristic polynomial
3. tr(Aⁱ)=tr(Bⁱ)

In this frame, we can use of course the adjacency matrix but also the Laplacian matrix L=D-A and the signless Laplacian matrix Q=D+A. For regular graphs, as far as cospectrality is concerned, there is no difference between the matrices A, L and Q. There is for example a direct relation between the eigenvalues of the signless Laplacian matrix of a graph, and the adjacency of its line graph. Following Sachs, let’s consider a connected graph G with n vertices and m edges. Let N the vertex-edge incidence matrix of G. Then rank(N)=n-1 if G is bipartite and rank(N)=n otherwise. In fact, we obtain for NN^T=Q and N^TN=2I+B, where Q is the signless Laplacian matrix of G and B the adjacency matrix of the line graph L(G) of G. NN^T and N^TN have the same non-zero eigenvalues with their multiplicities. So the spectrum of Q follows from the spectrum of Q and vice-versa. Classical examples of graphs determined by their spectrum are the complete graph, the regular complete bipartite graph and the cycle. A more complicate case can be found with d ≥3 with distance-regularity.This is the theorem of van Dam and Haemers.
If G is the distance-regular graph with diameter d and girth g satisfying one of the following properties, then every graph spectral with G is also distance-regular, with the same parameters as G.

1. g≥2d-1
2. g≥2d-2 and G is bipartite
3. g≥2d-2 and C_d-1C_d<-(C_d-1+1)(λ₁+ λ_d)
4. G is a generalized odd graph that is, α₁=…= α_d-1=0, α_d<>0
5. C₁=…= C_d=1
6. G is dodecahedron, or the isocahedron
7. G is the coset graph of the extended ternary Golay code
8. G is the Ivanov-Ivanov-Faradjev graph