Expansion in Matrix-Weighted Graphs

TL;DR

This paper introduces the concept of matrix-weighted graphs, extending classical graph theory notions like expansion, and establishes fundamental inequalities, opening avenues for constructing graphs with superior expansion properties.

Contribution

It generalizes the Laplacian and adjacency matrices to matrix-weighted graphs and proves analogues of key inequalities, suggesting new types of expanders with enhanced expansion.

Findings

Analogues of the expander mixing lemma hold for matrix-weighted graphs

A Cheeger-type inequality is established for these graphs

A new definition of matrix-weighted expanders is proposed

Abstract

A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to define and control expansion for matrix-weighted graphs. In particular, an analogue of the expander mixing lemma and one half of a Cheeger-type inequality hold for matrix-weighted graphs. A new definition of a matrix-weighted expander graph suggests the tantalizing possibility of families of matrix-weighted graphs with better-than-Ramanujan expansion.