On Pareto eigenvalue of distance matrix of graphs

TL;DR

This paper investigates Pareto eigenvalues of the distance matrix in connected graphs, revealing properties of eigenvectors, bounds on eigenvalues, and identifying graphs with optimal eigenvalues.

Contribution

It introduces new properties of Pareto eigenvectors in trees, establishes bounds for eigenvalues, and characterizes graphs with extremal second largest eigenvalues.

Findings

Non-zero entries of Pareto eigenvectors form a convex function in trees.

Minimum possible number of distance Pareto eigenvalues in connected graphs identified.

Lower bounds for the largest eigenvalues of the distance matrix established.

Abstract

In this article, we study Pareto eigenvalues of distance matrix of connected graphs and show that the non zero entries of every distance Pareto eigenvector of a tree forms a strictly convex function on the forest generated by the vertices corresponding to the non zero entries of the vector. Besides we find the minimum number of possible distance Pareto eigenvalue of a connected graph and establish lower bounds for $n$ largest distance Pareto eigenvalues of a connected graph of order $n.$ Finally, we discuss some bounds for the second largest distance Pareto eigenvalue and find graphs with optimal second largest distance Pareto eigenvalue.