Spectral Radius and Edge‐Disjoint Spanning Trees of Graphs With Prescribed Edge Connectivity

ABSTRACT Let and denote the edge connectivity and the maximum number of edge‐disjoint spanning trees contained in a graph , respectively. Fan, Gu, and Lin studied a tight spectral condition of a connected graph to guarantee . The famous theorem of Tutte and Nash‐Williams implies that and are closely related with , so it is interesting to explore conditions on a graph with to warrant . Since graphs with edge‐disjoint spanning trees are ‐edge‐connected, the converse does not hold. We initially provide a tight spectral radius condition to guarantee in ‐edge‐connected graphs with a unique extremal graph characterization. Moreover, an extensive spectral condition on in ‐edge‐connected graphs with fixed minimum degree is determined. For , we further study analogous results on 3‐edge‐connected graphs to ensure , which implies comprehensive spectral arguments on for .