Some results on the distance spectral radius and edge-disjoint spanning trees of graphs

Abstract

Let $\tau(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $\rho_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31--40] obtained a sharp distance spectral radius condition for $\tau(G)\ge k$ when $n\ge2k+6$. In this paper, we fill the gap $2k\le n\le2k+5$ and thus complete the result for all $n\ge2k$. The extremal graph given by Fan, He and Zhao remains valid for $n\ge2k+2$, while we determine the unique extremal graph for each of the orders $n=2k$ and $n=2k+1$. We further obtain sharp distance spectral radius conditions and characterize all extremal graphs under the minimum degree condition $\delta(G)\ge k$ for all $n\ge2k$. Finally, for graphs with the stronger minimum degree condition $\delta(G)\ge6k-4$ and order $n\ge2\delta(G)+2$, we obtain a sharp distance spectral radius condition ensuring $\tau(G)\ge k$ and determine the unique extremal graph.