An improved result on Laplacian spectral ratio of connected graphs

The Laplacian spectral ratio of a connected graph G, denoted by r(L)(G), is defined as the quotient between the largest and second smallest Laplacian eigenvalues of G. In 2002, Barahona and Pecora showed that rL(G) play an important role in the network synchronization control. In this paper, we obtain a result on the Laplacian spectral ratio of trees, which improve the known result of You and Liu [Z. You, B. Liu, On the Laplacian spectral ratio of connected graphs, Appl. Math. Lett. 25(2012)1245-1250]. Moreover, some graph operations on Laplacian spectral ratio arc given.