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For a simple graph  G , the signless Laplacian matrix of  G , denoted by  Q ( G ), is defined as  D ( G ) +  A ( G ), where  A ( G ) and  D ( G ) are the adjacency matrix and the diagonal matrix of vertex degrees of  G , respectively. By the smallest signless Laplacian eigenvalue of  G , denoted by  q ′( G ), we mean the smallest eigenvalue of  Q ( G ). Let  T ( n ,  t ) be the Turán graph on  n  vertices and  t  parts. In De Lima et al. (The clique number and the smallest Q-eigenvalue of graphs. Discrete Math. 2016;339:1744–1752), the authors posed the following conjecture: Conjecture.  Let  t  ≥ 3 and let  n  be sufficiently large. If  G  is a  Kt +1 -free graph of order  n  and  G  ≠  T ( n ,  t ), then  q ′( G ) &lt;  q ′( T ( n ,  t )).In this paper, first we disprove the above conjecture and then pose a new conjecture.

On a conjecture related to the smallest signless Laplacian eigenvalue of graphs