Unicyclic And Bicyclic Graphs With Extremal Lanzhou Index

Very recently D.Vukicevic et al.introduced a new topological index for a molecular graph G named the Lanzhou index as Lz(G)=?u?V(G)dudu2,where d,and du denote the degree of vertex u in G and in its complement respectively.The Lanzhou index Lz(G)can be expressed as(n-1)M1(G)-FG),where M1(G)and F(G)denote the first Zagreb index and the forgotten index of G.It turns out that the Lanzhou index outperforms M1(G)and F(G)in predicting the octanol-water partition coefficient for octane and nonane isomers.It was shown that stars and balanced double stars are the minimal and maximal trees for the Lanzhou index.In the article we reduce extremal problem of all graphs into one of a smaller class of graphs with some transformations.Then we obtain the minimum and maximum values for the Lanzhou index and characterize the corresponding extremal graphs.More precisely.For unicyclic graphs,the cycle length is 3 for all extremal unicyclic graphs except for some maximal graphs with 4?n?10.For bicyclic graphs,lengths of the two fundamental cycles(if the number of cycles are 3,we choose two cycles with smaller lengths as funda-mental cycles)are 3 for all extremal bicyclic graphs except for some maximal graphs with 5?n?9.In addition,for chemical graphs with maximum degree at most 4 and 3,we also obtain the minimum and maximum values as well as characterize the corresponding extremal graphs.Finally,we give the relationships between extremal graphs of unicyclic and bicyclic graphs.