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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...
Krishnamoorthy et al. [Minimum order graphs with specified diameter, connectivity and regularity, Networks 19 (1989) 25–46.] showed that a δ-regular graph with diameter D at most 3 has (vertex-)connectivity κ at least 2, and if D⩽2 then the connectivity is at least κ⩾min{δ,3}. Likewise, Soneoka et al [Sufficient conditions for maximally connected dense graphs, Discrete Math. 63 (1) (1987) 53–66] proved...
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