## 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⋅i with corresponding eigenspaces U_{i}(n,q) for 0≤i≤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)⊕U_{i+1}(n,q)⊕⋯⊕U_{j}(n,q) for 0≤i≤j≤n. For the case i+j≤n and q≥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 i+j>n and q≥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, i>[Formula presented] and q≥5. In particular, we characterize eigenfunctions from the eigenspace U_{i}(n,q) with the minimum cardinality of the support for cases i≤[Formula presented], q≥3 and i>[Formula presented], q≥5.

Original language | English |
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Pages (from-to) | 1351-1360 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 342 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 May 2019 |

## Keywords

- Eigenfunction
- Hamming graph
- Support
- CARDINALITY
- EIGENFUNCTIONS