Expander independent sets coloring and diameter

For $S\mathrm{\ne }\mathrm{\varnothing }$, mixing with $S=T$ gives $d\mathrm{\mid }S{\mathrm{\mid }}^2\mathrm{/}n\le \alpha d\mathrm{\mid }S\mathrm{\mid }(1-\mathrm{\mid }S\mathrm{\mid }\mathrm{/}n)$. Cancel the positive $d\mathrm{\mid }S\mathrm{\mid }$ and rearrange. For empty $S$ the bound holds directly. If $\alpha =0$ there is no nonempty independent set. In a loopless graph every color class is independent, so summing their sizes gives the color bound when $\alpha >0$.

Every set of size at most $n\mathrm{/}2$ has at least $h$ times its size in external neighbors, by the edge/vertex comparison. A ball therefore grows by a factor at least $1+h$ until it exceeds $n\mathrm{/}2$. With $R=\lceil \log n\mathrm{/}\log (1+h)\rceil +1$, a ball of radius $R$ must exceed half the graph; otherwise successive growth from its initial single vertex contradicts its size bound. Two such balls intersect, giving distance at most $2R$. Positive $h$ also excludes a separate component of size at most half. For $n=1$ use diameter zero without defining $h$.