Degree-ordered independent-set entropy bound ($i(G)\le {\prod }_i(2^{b_i+1}-1{)}^{1/d_i}$)

Order vertices by nonincreasing positive degrees $d_i$, and let $b_i$ count preceding neighbors. The Madiman-Tetali inequality applied to preceding-neighbor sets and $b_i$ copies of singleton $i$ bounds the entropy of a uniformly chosen independent set. In each local term, all-zero neighbor assignment permits two choices at $i$, while every other assignment permits at most one. The maximum-entropy bound gives ${\log }_2(2^{b_i+1}-1)$ for that term. Complete bipartite graphs $K_{d,d}$ attain equality when one part is ordered before the other.