The Wiener index, due to its many applications is considered to be one of very important distance-based index. But the Padmaker-Ivan (PI) index is kind of the only distance related index linked to parallelism of edges. The PI index like other distance related indices has great disseminating power. The index was firstly investigated by Khadikar et al. (2001), they have probed the chemical applications of the PI index. They proved that the proposed PI index correlates highly with the physicochemical properties and biological activities of a large number of diverse and complex chemical compounds and the Wiener and Szeged indices. Recently, the vertex Padmarkar-Ivan (PI v ) index of a chemical graph G was introduced as the sum over all edges uv of a molecular graph G of the vertices of the graph that are not equidistant to the vertices u and v . In this paper, the vertex PI v index of certain triangular tessellation are computed by using graph-theoretic analysis, combinatorial computing, and edge-dividing technology.

中文翻译：

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关于某些三角镶嵌网络的顶点PI指数

维纳指数，由于其应用广泛，被认为是非常重要的基于距离的指数之一。但是 Padmaker-Ivan (PI) 指数是唯一与边缘平行度相关的距离相关指数。PI指数与其他距离相关指数一样，具有很强的传播力。Khadikar 等人首先研究了该指数。(2001)，他们探索了 PI 指数的化学应用。他们证明，所提出的 PI 指数与大量不同且复杂的化合物的理化性质和生物活性以及 Wiener 和 Szeged 指数高度相关。最近，化学图 G 的顶点 Padmarkar-Ivan (PI v ) 索引被引入作为分子图 G 的所有边 uv 的总和，该图的顶点与顶点 u 和 v 不等距。