Maxima Q-index of graphs with forbidden odd cycles

Abstract

Let $q\left(G\right)$ be the $Q$-index (the largest eigenvalue of the signless Laplacian) of $G$. Let $S_{n,k}$ be the graph obtained by joining each vertex of a complete graph of order $k$ to each vertex of an independent set of order $n-k.$ The main result of this paper is the following theorem: Let $k\ge 3,$ $n\ge 110k^2,$ and $G$ be a graph of order $n$. If $G$ has no $C_{2k+1},$ then $q\left(G\right)<q\left(S_{n,k}\right),$ unless $G=S_{n,k}.$ This result proves the odd case of the conjecture in [M.A.A. de Freitas, V. Nikiforov, and L. Patuzzi, Maxima of the $Q$-index: forbidden $4$-cycle and $5$-cycle, \emph{Electron. J. Linear Algebra }26 (2013), 905-916.]

The new version contains a new proof of Lemma 10, as the previous one had a gap, which was pointed out by Dr. Bo Ning

Cited by in corpus (1)

• Maxima of the Q-index: forbidden even cycles