On the Conjecture for Certain Laplacian Integral Spectrum of Graphs

Let G be a simple graph of order n with Laplacian spectrum {lambda(n), lambda(n-1), ... , lambda(1)} where 0=lambda(n) <= lambda(n-1) <= ... <= lambda(1). 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 (mod 4). In this article, we have proved that (i) if G is connected and lambda(1) = n-1 then G has diameter either 2 or 3, and (ii) if lambda(1) = n-1 and lambda(n-1)= 1 then both G and (G) over bar, the complement of G, have diameter 3. (C) 2009 Wiley Periodicals, Inc. J Graph Theory 63: 106-113, 2010