On graphs with near minimal distance energy

Let G be a connected graph of order n   and let D=D(G)D=D(G) be its distance matrix. Suppose that λ1(D)≥⋯≥λn(D)λ1(D)≥⋯≥λn(D) are the eigenvalues of D(G)D(G). Then DE(G)=∑i=1n|λi(D(G))| is called the distance energy of G  . In this paper, we characterize all connected graphs with DE(G)∈[2n−2,2n]DE(G)∈[2n−2,2n].