The unique extremal graph maximizing edges or spectral radius among n-vertex graphs with b(G) < 1/r is characterized for each integer r ≥ 1, along with the bipartite case.
Qian, Two sufficient conditions for the existence ofk-factors in bipartite graphs, Shandong Daxue Xuebao Ziran Kexue Ban36(2001) 477-480
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Extremal results for graphs with binding number strictly less than $1/r$
The unique extremal graph maximizing edges or spectral radius among n-vertex graphs with b(G) < 1/r is characterized for each integer r ≥ 1, along with the bipartite case.