On a conjecture on a laplacian matrix with distinct integral spectrum

In a paper Fallat et al. (J Graph Theory 50 (2005), 162-174) consider the question of the existence of simple graphs G on n vertices whose Laplacian matrix L(G) has an integral spectrum consisting of simple eigenvalues only in the range 0,.,n, 0 always being, automatically, one of the eigenvalues. They completely characterize the case when n is one of the eigenvalues, but for the case when n is not, they conjecture that there are no such graphs. In that paper it is shown that, indeed, there are no such graphs for n≤11. In this paper we show that the conjecture is true for n≥6,649,688,933. We actually consider the nonexistence of graphs whose Laplacians are realized by more general spectra λ1≥λ2≥â̄≥λn, with λ1=n-1, λ2≤n-2, λn-2≥2, λn-1=1, and λn=0, subject to certain trace conditions. We show that, indeed, for sufficiently large n such graphs do not exist. Our methods are both graph theoretical and algebraic. In certain cases we refine the Cauchy interlacing theorem. Finally, rather than work with Laplacians which have nonpositive off-Diagonal entries, we transform the problems to the realizability of spectra of nonnegative matrices which we term anti-Laplacians.