# A coupling of the spectral measures at a vertex

## TL;DR

This paper introduces a joint spectral measure coupling for graph vertices based on adjacency matrices, enabling the recovery of matrix minors and extending a central limit theorem to multivariate cases.

## Contribution

It presents a novel coupling of spectral measures at vertices, extending Obata's CLT to multivariate star-graphs and exploring combinatorial properties via Viennot's heaps.

## Key findings

- Joint spectral measure captures rooted closed paths
- Allows recovery of minors of functions of the adjacency matrix
- Extends Obata's CLT to multivariate star-graphs

## Abstract

Given the adjacency matrix of an undirected graph, we define a coupling of the spectral measures at the vertices, whose moments count the rooted closed paths in the graph. The resulting joint spectral measure verifies numerous interesting properties that allow to recover minors of analytical functions of the adjacency matrix from its generalized moments. We prove an extension of Obata's Central Limit Theorem in growing star-graphs to the multivariate case and discuss some combinatorial properties using Viennot's heaps of pieces point of view.