Completely independent spanning trees and spectral radii of graphs

Let 𝑇 1 , 𝑇 2 , … , 𝑇 𝑘 be k spanning trees in a graph G . If for any two vertices 𝑢 , 𝑣 of G , the paths from u to v in 𝑇 1 , 𝑇 2 , … , 𝑇 𝑘 are pairwise edge-disjoint and internal vertex-disjoint, then 𝑇 1 , 𝑇 2 , … , 𝑇 𝑘 are k completely independent spanning trees (CISTs) of G . The concept of CISTs was first introduced by Hasunuma in 2001. In 2002, Hasunuma further proved that determining whether a graph G contains k CISTs is NP-complete, even for 𝑘 = 2 . In 2014, Araki raised the question of whether certain known sufficient conditions for Hamiltonian cycles could also guarantee the existence of two CISTs. Inspired by Araki's question and building upon spectral radius-based studies of Hamiltonicity in general graphs, we establish a spectral condition for the existence of CISTs in connected graphs for 𝑘 = 2 and 𝑘 = 3 .