On the sum of $k$-th largest distance eigenvalues of graphs

For a connected graph $G$ with order $n$ and an integer $k\geq 1$, we denote by $$S_k(D(G))=\lambda_1(D(G))+\cdots+\lambda_k(D(G))$$ the sum of $k$ largest distance eigenvalues of $G$. In this paper, we consider the sharp upper bound and lower bound of $S_k(D(G))$. We determine the sharp lower bounds of $S_k(D(G))$ when $G$ is connected graph and is a tree, respectively, and characterize both the extremal graphs. Moreover, we conjecture that the upper bound is attained when $G$ is a path of order $n$ and prove some partial result supporting the conjecture. To prove our result, we obtain a sharp upper bound of $\lambda_2(D(G))$ in terms of the order and the diameter of $G$, where $\lambda_2(D(G))$ is the second largest distance eigenvalue of $G$. As applications, we prove a general inequality involving $\lambda_2(D(G))$, the independence number of $G$, and the number of triangles in $G$. An immediate corollary is a conjecture of Fajtlowicz, which was confirmed in \cite{L15-L} by a different argument. We conclude this paper with some open problems for further study.