The Research On The PI Index Of Graphs

The Padmakar-Ivan (PI) index is a Wiener-Szeged-like topological index which reflects certain structural features of organic molecules. It has played an important role in describing molecular graph and establishing the relationship between the structure of molecule and its features. At the same time, PI index was extensively used to predict the physical and chemical properties of compounds and their biological activity. The PI index of a connected graph G is defined as follows: PI=PI(G)=Σ[neu(e|G)+nev(e|G)], where for edgee=(u,v), neu(e|G) is the number of edges of G lying closer to u than v, nev(e|G) is the number of edges of G lying closer to v than u and the summation goes over all edges of G. The edges which are equidistant from u and v are not considered for the calculation of PI index.We mainly consider three aspects about the research of PI index:The first is the research of some special forms of graphs such as product graph, chain graph, interval graph, strong string graph, outerplaner graph, Halin graph etc. The second is the re-search of the relationship between the PI index of original graph and it’s line graph, dual graph, imitative dual graph etc. The third is discussing methods and finding good algorithms which are suitable for calculating PI index. A series of researches have been done around these three aspects and had made grate advances with researcher’s further effort.On the basis of former researches, we get three results around the above mentioned three aspects using level sequence method and small graphs gluing into big graphs method.(1) The PI Index of the Mycielski’s Graph and the Generalized Mycielski’s Graph of Complete Graph(2) The relationship between the PI index of tree and it’s line graph(3) The PI index of the deficient sunflowers attached with lollipops.