Bell Numbers and Stirling Numbers of the Mycielskian of Trees

We establish explicit formulas for Bell numbers and graphical Stirling numbers of complete multipartite graphs, complete bipartite graphs with removed perfect matchings, and Mycielskian trees. For complete multipartite graphs $K(n_1,\ldots,n_\ell)$, we provide a simplified proof that $B(G) = \prod_{i=1}^\ell \bell{n_i}$. We derive $B(K_{n,n} - M) = \sum_{k=0}^{n} \binom{n}{k} \bell{k}^2$ for removed perfect matching $M$, and for Mycielskian star graphs, $B(M(St_n); 3) = 2^n + 1$ and $B(M(St_n); 2n) = 2n^2 - 3n + 3$. Results extend to Mycielskians of arbitrary trees. Our computational verifications establish links between graphical Bell numbers and fundamental sequences in combinatorics and pattern avoidance, including identification of several OEIS entries: A000051, A096376, A116735, A384980, A384981, A384988, A385432, and A385437.