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

TL;DR

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

Contribution

It introduces new spectral criteria involving the signless Laplacian index for detecting doubly chorded cycles in graphs, extending previous graph spectral theory.

Findings

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

Identifies specific graphs that are exceptions to the cycle containment.

Provides bounds on the index related to cycle existence.

Abstract

In this paper, we show that for a graph of order $n$, where $n\ge 5$, if the signless Laplacian index is larger than or equal to certain value depending on $n$, then the graph contains a doubly chorded cycle, where the chords incident to a common vertex, unless it is two specified graphs.