A conjecture on the diameter and signless Laplacian index of graphs

A bug Bugp,q1,q2Bugp,q1,q2 is a graph obtained from a complete graph KpKp by deleting an edge uv   and attaching paths Pq1Pq1 and Pq2Pq2 at u and v, respectively. In this paper, we show that for connected graphs G of order n   with signless Laplacian index q1(G)q1(G) and diameter diam(G)diam(G), q1(G)⋅diam(G)q1(G)⋅diam(G) is maximized for and only for the graph Bug⌊n/2⌋+2,p,qBug⌊n/2⌋+2,p,q, where p=⌊d/2⌋p=⌊d/2⌋, q=⌈d/2⌉q=⌈d/2⌉ and d=⌊(n+1)/2⌋d=⌊(n+1)/2⌋. This solves a conjecture in [6] on the signless Laplacian index involving the diameter.