The study aimed to obtain relationships between the omega invariants of a graph and its complement. We used some graph parameters, including the cyclomatic numbers, number of components, maximum number of components, order, and size of both graphs G and G. Also, we used triangular numbers to obtain our results related to the cyclomatic numbers and omega invariants of G and G. Several bounds for the above graph parameters have been obtained by the direct application of the omega invariant. We used combinatorial and graph theoretical methods to study formulae, relations, and bounds on the omega invariant, the number of faces, and the number of compo-nents of all realizations of a given degree sequence. Especially so-called Nordhaus-Gaddum type resulted in our calculations. In these calculations, the triangular numbers less than a given number play an important role. Quadratic equations and inequalities are intensively used. Several relations between the size and order of the graph have been utilized in this study. In this paper, we have obtained relationships between the omega invariants of a graph and its complement in terms of several graph parameters, such as the cyclomatic numbers, number of components, maximum number of components, order, and size of G and G, and triangular numbers. Some relationships between the omega invariants of a graph and its complement have been obtained.

中文翻译：

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补图的 Omega 不变量和 Nordhaus-Gaddum 类型结果

该研究旨在获得图的欧米伽不变量与其补集之间的关系。我们使用了一些图参数，包括图 G 和 G 的圈数、分量数、最大分量数、阶数和大小。此外，我们使用三角数来获得与圈数和 omega 不变量相关的结果G 和 G。上述图形参数的几个界限已通过直接应用 omega 不变量获得。我们使用组合和图论方法来研究欧米伽不变量的公式、关系和界限、面数以及给定度数序列的所有实现的分量数。特别是所谓的 Nordhaus-Gaddum 类型导致了我们的计算。在这些计算中，小于给定数的三角数起着重要作用。二次方程和不等式被大量使用。本研究中利用了图的大小和顺序之间的几种关系。在本文中，我们根据几个图参数（例如圈数、分量数、最大分量数、阶数和 G 和 G 的大小）获得了图的 omega 不变量与其补集之间的关系，并且三角形数字。已经获得了图的欧米伽不变量与其补集之间的一些关系。