Lattices Associated with Hamming Graphs

Abstract

Hamming graph \(H(n, k)\) has as vertex set all words of length \(n\) with symbols taken from a set of \(k\) elements. Suppose \(L\) denotes the set \(\bigcup_{i=0}^{n+1}\Omega_l\) with \(\Omega_l=\{\sum\limits_{i\in I_1}e_i^1+\sum\limits_{i\in I_2}e_i^2+\ldots+\sum\limits_{i\in I_k}e_i^k|I_j\cap I_j’=\emptyset (j\neq j’),|\bigcup_{j=1}^kI_j|=l\}\) for \(0\leq l\leq n\) and \(\Omega_{n+1}\). For any two elements \(x, y \in L\), define \(x \leq y\) if and only if \(y = I\) or \(I^x_j \leq I^y_j\) for some \(1 \leq j \leq k\). Then \(L\) is a lattice, denoted by \(L_o\). Reversing the above partial order, we obtain the dual of \(L_o\), denoted by \(L_r\). This article discusses their geometric properties and computes their characteristic polynomials.