In this work, we studied the problem of determining the values of the Zagreb indices of all the realizations of a given degree sequence. Methods: We first obtained some new relations between the first and second Zagreb indices and the forgotten index sometimes called the third Zagreb index. These relations also include the triangular numbers, order, size, and the biggest vertex degree of a given graph. As the first Zagreb index and the forgotten index of all the realizations of a given degree sequence are fixed, we concentrated on the values of the second Zagreb index and studied several properties including the effect of vertex addition. Results: In our calculations, we make use of a new graph invariant, called omega invariant, to reach numerical and topological values claimed in the theorems. This invariant is closely related to Euler char-acteristic and the cyclomatic number of graphs. Conclusion: Therefore this invariant is used in the calculation of some parameters of the molecular structure under review in terms of vertex degrees, eccentricity, and distance.

中文翻译：

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萨格勒布指数与 Omega 不变量之间的新关系

在这项工作中，我们研究了确定给定度序列的所有实现的萨格勒布指数值的问题。方法：我们首先获得第一和第二萨格勒布索引与被遗忘的索引（有时称为第三萨格勒布索引）之间的一些新关系。这些关系还包括给定图的三角形数、阶数、大小和最大顶点度数。由于第一个 Zagreb 索引和给定度数序列的所有实现的遗忘索引都是固定的，因此我们专注于第二个 Zagreb 索引的值，并研究了包括顶点相加效果在内的几个属性。结果：在我们的计算中，我们利用一种新的图不变量（称为 omega 不变量）来达到定理中要求的数值和拓扑值。这个不变量与欧拉特性和图的圈数密切相关。结论：因此，该不变量可用于计算所审查的分子结构的一些参数，如顶点度、偏心率和距离。