Haemers rank bound ($c(G)\le {\mathrm{rank}}_{F}A$)

A fitting matrix over a field has nonzero diagonal entries and zero entries at distinct nonadjacent vertices. Its tensor powers fit strong graph powers. On an independent set the restricted matrix is diagonal and nonsingular, so its cardinality is at most the matrix rank. This yields the capacity upper bound over every field.

Choose points $a_v\in F^r$ and polynomials $f_v$ in a finite-dimensional polynomial space $M$, with $f_v(a_v)\ne 0$ and $f_v(a_w)=0$ for distinct nonadjacent vertices. Evaluation produces a fitting matrix of rank at most ${\dim }_{F}M$. Equivalently, tensoring the polynomial spaces gives linearly independent polynomials for every independent code in a strong power, proving $c(G)\le {\dim }_{F}M$.