A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. Our results improve a bound of Lee, Loh, and Sudakov (J. Comb. Th. Ser. B 103 (2013)) for (unweighted) maximum bisections in graphs whose maximum degree is either even or equals 3, and for almost all graphs. We show that a tight lower bound for maximum size of bisections in 3-regular graphs obtained by Bollob\'as and Scott (J. Graph Th. 46 (2004)) can be extended to weighted subcubic graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs especially if we exclude $K_{1,3}$. We pose three conjectures.

翻译：二分图是指将图的顶点分成两部分，且两部分顶点数相差不超过1的切割。本文研究了最大度有界边赋权图中最大权重二分的下界。我们的结果改进了Lee、Loh和Sudakov（J. Comb. Th. Ser. B 103 (2013)）关于（非赋权）图最大二分的边界，这些图的最大度要么为偶数，要么等于3，且适用于几乎所有图。我们证明，Bollobás和Scott（J. Graph Th. 46 (2004)）针对3-正则图得到的最大二分的紧下界可以推广到赋权次立方图。此外，我们考虑边赋权无三角形次立方图，并表明对于此类图（尤其是排除$K_{1,3}$的情况），比边赋权次立方图更优的下界成立。我们提出了三个猜想。