On the Smallest Signless Laplacian Eigenvalue of A Graph

The signless laplacian matrix of a graph is defined as the summation of the adjacency matrix andthe diagonal matrix of the vertex degree of the graph. we study the smallest eigenvalue of the signlesslaplacian matrix of graphs. we investigate some conjectures related the smallest signless laplacianeigenvalues of graphs. using our results, we disprove one of those conjectures.