page 10 : maximum leaf number $k$ upper bounds bandwidth by $\mathcal O(k)$ – Lemma 4.25. The max leaf number $\mathrm{ml}$ strictly upper bounds the bandwidth $\mathrm{bw}$.
page 22 : bandwidth – Let $G=(V,E)$ be a graph, and let $f\colon V\to {1,2,\dots,n}$ be a linear ordering of $G$. 1. The \emph{bandwidth} of $f$ is $\max{|f(v)-f(w)| \mid (v,w) \in E}$. … The bandwidth … is the minimum bandwidth … over all possible linear orderings of $G$.
page 23 : bandwidth $k$ upper bounds pathwidth by $\mathcal O(k)$ – Theorem 44. For every graph $G$, the pathwidth of $G$ is at most the bandwidth of $G$, … Proof. Let $f \colon V\to {1,\dots,n}$ be a linear ordering of $G$ with bandwidth $k$. Then $(X_1,\dots,X_{n-k})$ with $X_i={f^{-1}(i), f^{-1}(i+1), \dots, f^{-1}(i+k)}$ is a path decomposition of $G$ with pathwidth $k$. …
bandwidth – (paraphrased) Label graph vertices with distinct integers. Bandwidth of this labelling is the maximum over label differences over all edges. Bandwidth of a graph is the minimum over all labellings.
bandwidth $k$ upper bounds maximum degree by $\mathcal O(k)$ – Each vertex has an integer $i$ and may be connected only to vertices whose difference from $i$ is at most $k$. There are at most $k$ bigger and $k$ smaller such neighbors.
bandwidth $k$ implies that cutwidth is $k^{\mathcal O(1)}$ – Any bandwidth bound cutwidth quadratically. An example where this happens is $(P_n)^k$ which has bandwidth $k$ and cutwidth $O(k^2)$; both seem to be optimal.
