Maxima of the q-index: Graphs without long paths

This paper gives tight upper bound on the largest eigenvalue q (G) of the signless Laplacian of graphs with no paths of given order. Thus, let Sn, k be the join of a complete graph of order k and an independent set of order n - k, and let S+n,k be the graph obtained by adding an edge to Sn, k The main result of the paper is the following theorem: The main ingredient of our proof is a stability result of its own interest, about graphs with large minimum degree and with no long paths. This result extends previous work of Ali and Staton