Combinatorics / CONCEPT

Ordinary zero forcing, symmetric maximum nullity, and minimum rank of unequal-leg spider powers

For integers m >= 2 and k >= 1 and independently chosen leg lengths ell_0, ..., ell_{m - 1} >= k + 1, the ordinary distance-at-most-k power G = T^k of the spider T on N = 1 + sum_{i = 0}^{m - 1} ell_i vertices has ordinary zero forcing number Z(G) = (m - 1) k, real symmetric maximum nullity M_R(G) = (m - 1) k, and real symmetric minimum rank mr_R(G) = N - (m - 1) k. The upper bound Z(G) <= (m - 1) k is realized by the tail set consisting of the last k vertices on each of the first m - 1 legs, while the matching lower bound and symmetric extremum are realized by one explicit real symmetric matrix combining per-leg sliding-window Gram blocks D_{ell_i}^T D_{ell_i} with cross-leg and center indicator couplings.