Graphs with degree sequence { ( m-1)m, m , (n n-1)n} n } and {mn, m n , n m }

In this paper we study the class of graphs G m , n that have the same degree sequence as two disjoint cliques Km m and K n , as well as the class G m , n of the complements of such graphs. We establish various properties of G m , n and G m , n related to recognition, connectivity, diameter, bipartiteness, Hamiltonicity, and pancyclicity. We also show that several classical optimization problems on these graphs are NP-hard, while others can be solved efficiently. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.