On the conjecture for certain laplacian integral spectrum of graphs

Let G be a simple graph of order n with Laplacian spectrum [γn, γn-1, . . .,γ1} where 0 = γn ≤ γ 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 γ1 = n-1 then G has diameter either 2 or 3, and (ii) if γ1 = n- 1and γn--1 = 1 then both G and Ḡ, the complement of G, have diameter 3-