Bandwidth of the product of paths of the same length

TL;DR

This paper provides a numerical expression for the bandwidth of the d-product of a path graph, showing it as a sum of multinomial coefficients and analyzing its bounds and asymptotic behavior.

Contribution

It introduces a precise formula for the bandwidth of the d-product of a path and compares its asymptotic behavior with lexicographic ordering.

Findings

Bandwidth expressed as sum of multinomial coefficients

Bounds established using coefficients from polynomial expansions

Asymptotic comparison with lexicographic ordering

Abstract

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.