# Maxima of the Q-index: Forbidden 4-cycle and 5-cycle

## Abstract

This paper gives tight upper bounds on the largest eigenvalue q (G) of the signless Laplacian of graphs with no 4-cycle and no 5-cycle. If n is odd, let Fn be the friendship graph of order n; if n is even, let Fn be Fn-1 with an extra edge hung to its center. It is shown that if G is a graph of order n ≥ 4, with no 4-cycle, then q (G) < q (Fn), unless G = Fn. Let Sn,k be the join of a complete graph of order k and an independent set of order n - k. It is shown that if G is a graph of order n ≥ 6, with no 5-cycle, then q (G) < q (Sn,2), unless G = Sn,k. It is shown that these results are significant in spectral extremal graph problems. Two conjectures are formulated for the maximum q (G) of graphs with forbidden cycles.

## Publication Title

Electronic Journal of Linear Algebra

## Recommended Citation

De Freitas, M., Nikiforov, V., & Patuzzi, L.
(2014). Maxima of the Q-index: Forbidden 4-cycle and 5-cycle.* Electronic Journal of Linear Algebra**, 26*, 905-916.
Retrieved from https://digitalcommons.memphis.edu/facpubs/5094