Gram matrices of reproducing kernel Hilbert spaces over graphs

TL;DR

This paper introduces a new framework for graph analysis using reproducing kernel Hilbert spaces and investigates the properties of their Gram matrices, providing bounds and characterizations for different graphs.

Contribution

It develops the concept of RKHS for graphs and analyzes the Gram matrices, offering bounds and characterizations that were not previously established.

Findings

Derived bounds on Gram matrix entries

Characterized graphs attaining these bounds

Provided theoretical insights into graph structures

Abstract

In this paper, we introduce the notion of reproducing kernel Hilbert spaces for graphs and the Gram matrices associated with them. Our aim is to investigate the Gram matrices of reproducing kernel Hilbert spaces. We provide several bounds on the entries of the Gram matrices of reproducing kernel Hilbert spaces and characterize the graphs which attain our bounds.