Background: The omega index has been recently introduced to identify a variety of topological and combinatorial aspects of a graph with a new viewpoint. As a continuing study of the omega index, by considering the incidence of edges and vertices to the adjacency of the vertices, in this paper, we have introduced the second omega index Ω2 and then computed it over some well-known graph classes. Methods: Many combinatorial counting methods have been utilized in the proofs. The edge partition is frequently used throughout the work. Naturally, some graph theoretical lemmas are also used. Results: In particular, trees having the smallest and largest Ω2 have been constructed. Finally, the second omega index of some derived graphs, such as line graphs, subdivision graphs, and vertex-semitotal graphs, has been presented. Conclusion: Omega invariant has already been explored in many papers. It has been defined in terms of vertex degrees. Vertices correspond to the atoms in a molecule and a calculation, which only depends on the atomic parameters, is not even comparable with a calculation containing both atoms and chemical bonds between them. With this idea in mind, we have evaluated some mathematical properties of the second omega index, which has great potential in chemical applications related to the number of cycles in the molecular graph.

中文翻译：

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第二个欧米茄指数

背景：最近引入了欧米伽索引，以一种新的观点来识别图的各种拓扑和组合方面。作为对 omega 指数的持续研究，通过考虑边和顶点与顶点邻接的关联，在本文中，我们引入了第二个 omega 指数 Ω2，然后在一些众所周知的图类上计算它。方法：证明中使用了许多组合计数方法。整个作品中频繁使用边缘隔断。当然，也使用了一些图论引理。结果：特别是，已经构建了具有最小和最大Ω2 的树。最后，提出了一些派生图（例如折线图、细分图和点半全图）的第二欧米伽索引。结论：Omega 不变量已经在许多论文中进行了探讨。它是根据顶点度数来定义的。顶点对应于分子中的原子，仅取决于原子参数的计算甚至无法与同时包含原子和原子之间的化学键的计算相比。考虑到这个想法，我们评估了第二个欧米伽指数的一些数学特性，它在与分子图中的循环数相关的化学应用中具有巨大的潜力。