Maxima of the Q-index: forbidden even cycles

TL;DR

This paper characterizes the maximum signless Laplacian eigenvalue among graphs of large order with forbidden even cycles, confirming a conjecture and identifying extremal graphs.

Contribution

It proves a conjecture by de Freitas, Nikiforov, and Patuzzi by determining the extremal graph for the maximum signless Laplacian eigenvalue under cycle length restrictions.

Findings

Identifies the extremal graph $S_{n,k}^{+}$ for the maximum $q$-index.

Shows that no other graph with the same cycle restrictions surpasses $S_{n,k}^{+}$ in $q$-index.

Completes the proof of a previously conjectured extremal spectral property.

Abstract

Let $G$ be a graph of order $n$ and let $q\left( G\right) $ be 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 $n-k;$ and let $S_{n,k}^{+}$ be the graph obtained by adding an edge to $S_{n,k}.$ It is shown that if $k\geq2,$ $n\geq400k^{2},$ and $G$ is a graph of order $n,$ with no cycle of length $2k+2,$ then $q\left( G\right) <q\left( S_{n,k}^{+}\right) ,$ unless $G=S_{n,k}^{+}.$ This result completes the proof of a conjecture of de Freitas, Nikiforov and Patuzzi.