Signless Laplacian index conditions for trebly chorded cycles in graphs with given order

TL;DR

This paper establishes conditions based on the signless Laplacian index that guarantee the presence of trebly chorded cycles in graphs of order at least six, with specific exceptions.

Contribution

It introduces new spectral conditions involving the signless Laplacian index that ensure the existence of trebly chorded cycles in graphs, identifying two exceptional graph cases.

Findings

Graphs with sufficiently large signless Laplacian index contain trebly chorded cycles.

Two specific graphs are exceptions to the cycle containment condition.

The results apply to graphs with order n ≥ 6.

Abstract

It is proved that for a graph of order $n$, where $n\ge 6$, if the signless Laplacian index is larger than or equal to certain value depending on $n$, then the graph contains a trebly chorded cycle, where the chords incident to a common vertex, unless it is one of two specified graphs.