Graph bandwidth

In graph theory, the graph bandwidth problem is to label the n vertices v_i of a graph G with distinct integers ${\displaystyle f(v_{i})}$ so that the quantity ${\displaystyle \max\{\,|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}}$ is minimized (E is the edge set of G). The problem may be visualized as placing the vertices of a graph at distinct integer points along the x-axis so that the length of the longest edge is minimized. Such placement is called linear graph arrangement, linear graph layout or linear graph placement.

The weighted graph bandwidth problem is a generalization wherein the edges are assigned weights w_ij and the cost function to be minimized is ${\displaystyle \max\{\,w_{ij}|f(v_{i})-f(v_{j})|:v_{i}v_{j}\in E\,\}}$.

In terms of matrices, the (unweighted) graph bandwidth is the minimal bandwidth of a symmetric matrix which is an adjacency matrix of the graph. The bandwidth may also be defined as one less than the maximum clique size in a proper interval supergraph of the given graph, chosen to minimize its clique size (Kaplan & Shamir 1996).