Maxima Q-index of graphs with forbidden odd cycles

TL;DR

This paper determines the maximum signless Laplacian eigenvalue for large graphs without odd cycles of a certain length, confirming a conjecture for the odd cycle case.

Contribution

It proves the conjecture that the maximum $Q$-index is achieved by a specific graph structure when odd cycles are forbidden.

Findings

Identifies the extremal graph $S_{n,k}$ for the $Q$-index under odd cycle restrictions.

Shows that for large $n$, the $Q$-index of any graph without $C_{2k+1}$ is less than that of $S_{n,k}$.

Validates the odd cycle case of a conjecture relating forbidden cycles and spectral extremal graphs.

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 $k\geq3,$ $n\geq110k^{2},$ 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.]