Spectral radius, edge-disjoint cycles and cycles of the same length

TL;DR

This paper establishes spectral conditions that ensure the existence of specific cycle configurations in graphs, such as edge-disjoint cycles and cycles of the same length, extending classical combinatorial results through spectral analysis.

Contribution

It introduces spectral criteria for cycle existence, including multiple edge-disjoint cycles and cycles of identical length, using eigenvector techniques.

Findings

Spectral conditions guarantee two edge-disjoint cycles.

Spectral conditions guarantee two cycles of the same length.

Spectral criteria ensure the existence of k edge-disjoint triangles.

Abstract

In this paper, we give spectral conditions to guarantee the existence of two edge disjoint cycles and two cycles of the same length. These two results can be seen as spectral analogues of Erd\H{o}s and Posa's size condition and Erd\H{o}s' classic problem on non existence of two cycles of the same length. By using double leading eigenvectors skill, we further give spectral condition to guarantee the existence of $k$ edge disjoint triangles.