Minimum supports of functions on the Hamming graphs with spectral constraints

TL;DR

This paper determines the minimum support size of functions on Hamming graphs with spectral constraints, providing characterizations for various parameter regimes and extending understanding of eigenfunctions with minimal nonzero entries.

Contribution

It introduces new bounds and characterizations for the minimum support of functions in specific eigenspaces of Hamming graphs, generalizing previous results.

Findings

Minimum support sizes are established for functions in certain eigenspaces.

Characterizations of functions with minimal support are provided for various parameter cases.

Results extend spectral support bounds to broader classes of Hamming graph functions.

Abstract

We study functions defined on the vertices of the Hamming graphs $H(n,q)$. The adjacency matrix of $H(n,q)$ has $n+1$ distinct eigenvalues $n(q-1)-q\cdot i$ with corresponding eigenspaces $U_{i}(n,q)$ for $0\leq i\leq n$. In this work, we consider the problem of finding the minimum possible support (the number of nonzeros) of functions belonging to a direct sum $U_i(n,q)\oplus U_{i+1}(n,q)\oplus\ldots\oplus U_j(n,q)$ for $0\leq i\leq j\leq n$. For the case $n\geq i+j$ and $q\geq 3$ we find the minimum cardinality of the support of such functions and obtain a characterization of functions with the minimum cardinality of the support. In the case $n<i+j$ and $q\geq 4$ we also find the minimum cardinality of the support of functions, and obtain a characterization of functions with the minimum cardinality of the support for $i=j$, $n<2i$ and $q\geq 5$. In particular, we characterize eigenfunctions from the eigenspace $U_{i}(n,q)$ with the minimum cardinality of the support for cases $i\le \frac{n}{2}$,$q\ge 3$ and $i> \frac{n}{2}$,$\,q\ge 5$.