On the signless Laplacian index and radius of graphs

A bag Bagp,q is a graph obtained from a complete graph Kp by replacing an edge uv by a path Pq. In this paper, we show that for all the connected graphs of order n≥5 with signless Laplacian index q1(G) and radius rad(G), q1(G)⋅rad(G) is maximum for and only for the graph Bagn−2s+3,2s−1, where s=⌈n/4⌉. This solves a conjecture in [6] on the signless Laplacian index involving the radius.