On the Edge Derivative of the Normalized Laplacian with Applications to Kemeny's Constant

TL;DR

This paper investigates how small changes in a graph's structure affect Kemeny's constant, a measure of connectivity, by deriving derivatives of the normalized Laplacian and establishing bounds for these derivatives.

Contribution

It introduces the directional derivative of Kemeny's constant and normalized Laplacian eigenvalues, providing new bounds and insights into graph connectivity changes.

Findings

Derived the directional derivative of Kemeny's constant for various graph families.

Established sharp bounds for the directional derivative of normalized Laplacian eigenvalues.

Provided bounds for the directional derivative of Kemeny's constant.

Abstract

In a connected graph, Kemeny's constant gives the expected time of a random walk from an arbitrary vertex $x$ to reach a randomly-chosen vertex $y$. Because of this, Kemeny's constant can be interpreted as a measure of how well a graph is connected. It is generally unknown how the addition or removal of edges affects Kemeny's constant. Inspired by the directional derivative of the normalized Laplacian, we derive the directional derivative of Kemeny's constant for several graph families. In addition, we find sharp bounds for the directional derivative of an eigenvalue of the normalized Laplacian and bounds for the directional derivative of Kemeny's constant.