A q-analog of the adjacency matrix of the n-cube

TL;DR

This paper introduces a q-analog of the adjacency matrix for the n-cube, computes its eigenvalues and eigenbasis, and counts rooted spanning trees, extending known algebraic structures with new combinatorial insights.

Contribution

It defines a new q-analog of the n-cube's adjacency matrix, determines its spectral properties, and provides a weighted enumeration of spanning trees, connecting algebraic and combinatorial aspects.

Findings

Eigenvalues and eigenbasis of the q-analog matrix are explicitly determined.

A weighted count of rooted spanning trees in the q-analog n-cube is provided.

The q-analog relates to known algebraic structures like Leonard pairs and Kac matrices.

Abstract

We define a q-analog of the adjacency matrix of the n-cube, determine its eigenvalues and write down a canonical eigenbasis. We give a weighted count of the number of rooted spanning trees in the q-analog of the n-cube. Remarks on the previous version: The q-analog of the Kac matrix appears in Terwilliger's classification of Leonard pairs and as such its eigenvalues and eigenvectors were known. Reference added to Terwilliger's papers and also to a paper of Johnson. Title changed to reflect this.