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.
Electronic Journal of Linear Algebra
Nikiforov, V., & Yuan, X. (2014). Maxima of the q-index: Graphs without long paths. Electronic Journal of Linear Algebra, 27, 504-514. https://doi.org/10.13001/1081-3810.1633