Bandwidth of the product of paths of the same length
  In this note we give a numerical expression for the bandwidth $bw(P_{n}^{d})$ of the $d$-product of a path with $n$ edges, $P_{n}^{d}$. We prove that this bandwidth is given by the sum of certain multinomial coefficients. We also show that $bw(P_{n}^{d})$ is bounded above and below by the largest coefficient in the expansion of $(1+x+...+x^{n})^{k}$, with $k\in{d,d+1}$. Moreover, we compare the asymptotic behavior of $bw(P_{n}^{d})$ with the bandwidth of the labeling obtained by ordering the vertices of $P_{n}^{d}$ in lexicographic order. 
Louis J. Billera and Sa\'ul A. Blanco
math.CO cs.DM