Bandwidth of the product of paths of the same length

In this note we give a numerical expression for the bandwidth bw(Pnd) of the dd-product of a path with nn edges, Pnd. We prove that this bandwidth is given by the sum of certain multinomial coefficients. We also show that bw(Pnd) is bounded above and below by the largest coefficient in the expansion of (1+x+⋯+xn)k(1+x+⋯+xn)k, with k∈{d,d+1}k∈{d,d+1}. Moreover, we compare the asymptotic behavior of bw(Pnd) with the bandwidth of the labeling obtained by ordering the vertices of Pnd in lexicographic order.