Abstract

The imbalance of an edge uv in a graph G is defined as |d(u)-d(v)|, where d(u) denotes the degree of u. The irregularity of G, denoted irr(G), is the sum of the edge imbalances taken over all edges in G. We determine the structure of bipartite graphs having maximum possible irregularity with given cardinalities of the partite sets and given number of edges. We then derive a corresponding result for bipartite graphs with given cardinalities of the partite sets and determine an upper bound on the irregularity of these graphs. In particular, we show that if G is a bipartite graph of order n with partite sets of equal cardinalities, then irr(G)n³/27, while if G is a bipartite graph with partite sets of cardinalities n₁ and n₂, where n₁2n₂, then irr(G)irr(K_n1,n2).

Keywords: Bipartite; Edge imbalance; Graph irregularity

Mathematical subject codes: 05C35

Article Outline

1. Introduction

2. The structure of extremal graphs

3. Extremal graphs with arbitrary size

4. Maximum values of the irregularity

Acknowledgements

References

Research supported in part by the University of Natal and the South African National Research Foundation.