On the irregularity strength of the <i>m</i> × <i>n</i> grid

Abstract

Abstract Given a graph G with weighting w : E ( G ) ← Z + , the Strength of G ( w ) is the maximum weight on any edge. The sum of a vertex in G ( w ) is the sum of the weights of all its incident edges. The network G ( w ) is irregular if the vertex sums are distinct. The irregularity strength of G is the minimum strength of the graph under all irregular weightings. In this paper we determine the irregularity strength of the m × n grid for certain m and n . In particular, for every positive integer d we find the irregularity strength for all but a finite number of m × n grids where n ‐ m = d . In addition, we present a general lower bound for the irregularity strength of graphs. © 1992 John Wiley &amp; Sons, Inc.