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

TL;DR

This paper investigates bounds on the sum of the top k distance eigenvalues of graphs, characterizes extremal cases, and supports a conjecture about the maximum being achieved by paths.

Contribution

It establishes sharp bounds for the sum of the largest distance eigenvalues, characterizes extremal graphs for the lower bounds, and provides partial results supporting a conjecture on the upper bound.

Findings

Determined sharp lower bounds for connected graphs and trees.

Proved a partial result supporting the conjecture that paths maximize the sum.

Derived bounds involving the second largest distance eigenvalue, graph diameter, and other parameters.

Abstract

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.