 Independent-set bound for biregular graphs

For a biregular graph with $m$ left vertices of degree $r$ and $n$ right vertices of degree $s$, let $X$ be the indicator vector of a uniformly selected independent set. Use base-two information entropy. Shearer inequality gives $H(X_U)\le r^{-1}{\sum }_{w\in W}H(X_{N(w)})$. Conditional on $X_U$, each right vertex with an empty selected neighbourhood is an independent fair binary choice, and the other right vertices are forbidden. Writing $q_w=\mathbb{P}(X_{N(w)}=0)$ gives $H(X)\le r^{-1}{\sum }_w[H(X_{N(w)})+r{q}_w]$. For the $2^s$ neighbourhood patterns, assign weight $2^r$ to the zero pattern and weight one to all others. Jensen's inequality implies $H(Y)+r{q}_w\le {\log }_2(2^r+2^s-1)$. Since $n/r=m/s$, the bound follows from $H(X)={\log }_{2}i(G)$. If $s$ divides $m$, the disjoint union of $m/s$ copies of the complete bipartite graph $K_{s,r}$ attains equality: each component has $2^s+2^r-1$ independent sets.