On the conjecture for certain Laplacian integral spectrum of graphs



Let G be a simple graph of order n with Laplacian spectrum {λn, λn−1, …, λ1} where 0=λn≤λn−1≤⋅≤λ1. If there exists a graph whose Laplacian spectrum is S={0, 1, …, n−1}, then we say that S is Laplacian realizable. In 6, Fallat et al. posed a conjecture that S is not Laplacian realizable for any n≥2 and showed that the conjecture holds for n≤11, n is prime, or n=2, 3(mod4). In this article, we have proved that (i) if G is connected and λ1=n−1 then G has diameter either 2 or 3, and (ii) if λ1=n−1 and λn−1=1 then both G and Ḡ, the complement of G, have diameter 3. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 106–113, 2010