Summary: Let \(D(G)=(d_{i,j})_{n\times n}\) denote the distance matrix of a connected graph \(G\) with order \(n\), where \(d_{ij}\) is equal to the distance between vertices \(v_{i}\) and \(v_{j}\) in G. The value \(D_{i}=\sum_{j=1}^{n}d_{ij}(i=1,2, \dots ,n\)) is called the distance degree of vertex \(v_{i}\). Denote by \(\rho (G),\rho_{n}(G)\) the largest eigenvalue and the smallest eigenvalue of \(D(G)\) respectively. The distance spectral spread of a graph \(G\) is defined to be \(S_{D}(G)=\rho (G) - \rho_{n}(G)\).
In this paper, some lower bounds on \(S_{D}(G)\) are given in terms of distance degrees, the largest vertex degree and clique number; the spreads of \(K_{n}\), \(K_{\lfloor \frac {1}{2}\rfloor,\lceil \frac {1}{2} \rceil}\), \(K_{n - \alpha }\nabla \alpha K_{1}\), \(K_{1,n - 1}\) are proved to be the least among all connected graphs with \(n\) vertices, all bipartite graphs with \(n\) vertices, all the graphs with both \(n\) vertices and independent number \(\alpha \), all trees with \(n\) vertices respectively.