Maxima of the Q-index: forbidden 4-cycle and 5-cycle

TL;DR

This paper establishes tight upper bounds on the maximum signless Laplacian eigenvalue for graphs excluding 4-cycles and 5-cycles, identifying extremal graphs and proposing related conjectures.

Contribution

It provides new spectral bounds for graphs without small cycles and characterizes extremal graphs achieving these bounds.

Findings

q(G)<q(F_{n}) for graphs with no 4-cycle, unless G=F_{n}

q(G)<q(S_{n,2}) for graphs with no 5-cycle, unless G=S_{n,k}

Formulation of two conjectures on maximum q(G) for graphs with forbidden cycles

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 F_{n} be the friendship graph of order n; if n is even, let F_{n} be F_{n-1} with an edge hanged to its center. It is shown that if G is a graph of order n, with no 4-cycle, then q(G)<q(F_{n}), unless G=F_{n}. Let S_{n,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, with no 5-cycle, then q(G)<q(S_{n,2}), unless G=S_{n,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.