Maxima of the Q-index: forbidden even cycles

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 $k\ge 2,$ $n\ge 400k^2,$ 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.

16 pages

Cited by in corpus (1)

• On the signless Laplacian spectral radius of $C_4$-free $k$-cyclic graphs