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The study of the topological indices of graphs derived from the combination of chemical topology and graph theory. Wiener index, Szeged index, Padmakar-Ivan index, and Sadhana index are several common topological indices. Wiener index, Szeged index, and Padmakar-Ivan index are all used to reflect certain structural features of organic molecules, and they all have very broad use in theoretical chemistry. Since Wiener and Szeged indices of acyclic graphs coincide, Padmakar proposed another topological index in2000, which he named Padmakar-Ivan index and abbreviated as PI, Padmakar conceived this index while attempting simultaneous estimation of Wiener and Szeged indices. The PI index of graph G is defined as follows:PI=PI(G)=Σe∈E(G)[neu(e|G)+nev(e|G)]. Here, we define edge of G connecting the vertices u and v, as e=uv∈E(G). The quan-tities neu(e|G) and nev(e|G) are the number of edges closer to u and v respectively. In calculating PI index edges equidistance from both end of the edge uv are not counted.About the PI indices of graphs, the predecessors have done a lot of meaningful results. Such as, the PI index of T with n vertices is (n-1)(n-2), the PI index of C2n+1is2n(n+1), the PI index of C2n is4n(n-1). And the PI indices of some typical organic molecules (polyacenes, helicencs, polyphenylenes, etc.) are obtained. Recently, the researchers mainly devote to study the PI index in nanotechnology, especially the PI indices of nanotubes, and they have got voluminous achievement. Such as, PI index is a uscfull tool for quantitative structure-activity-property-toxicity relationships(QSAR, QSPR, QSTR); calculate the PI indices of nanostructurcs, and so on.In this paper, we mainly studied the PI indices of the simple connected graphs. The structural features of Fans (Fn) and uniform inflations of Fans (UFFn) were studied, and we got the structures of Fans and uniform inflations of Fans had some symmetry, then the PI indices of Fans and uniform inflations of Fans(UFFn) were presented. Then, we presented the PI indices of the Halin graphs with given order and leaf number, and we characterized the Halin graphs with the corresponding PI indices. Lastly, by the method of systematic classification, we solved the PI indices with respect to the simple pericondensed hexagonal systems.
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