The Extremal Values Of Topological Index Of Caterpillars With Given Degree Sequence

Rand_i(?) ind_ex and Wiener ind_ex are two important molecular topological ind_ices in chemical graph theory. Moreover, in the general research about graph theory, they also have important theoretical significances. So in this paper we will stud_y the two ind_ices. In the paper, we will d_iscuss some properties of the extremal caterpillars which respectively attain the extremal values of Rand_i(?) ind_ex and Wiener ind_ex among all caterpillars with given d_egree sequence. And accord_ing to the properties, the structure of the extremal caterpillars will become clear.Firstly, we will use the method_s in algebra and graph theory to characterize the extremal caterpillar with the maximum/minimum Rand_i(?) ind_ex among all caterpillars with given d_egree sequenceÏ€=( d₁ , d 2, ..., d_n), where d₁ (?) d 2 ?...(?) d k (?) 2(?) d_k+1=...=d_n=1 Moreover, subsequently we will learn that when k is fixed the extremal caterpillar with the maximum/minimum Rand_i(?) ind_ex is unique. In ad_dition, we will point out the cond_ition"d₁ (?) d 2 ?...?d_k"is ind_ispensable.Second_ly, we will consid_er the Wiener ind_ex of the caterpillars with given d_egree sequenceÏ€=( d₁ , d 2, ..., d_n), where d₁ (?) d 2 ?...(?) d k (?) 2(?) d_k+1=...=d_n=1 and characterize the extremal caterpillars which respectively attain the maximum and minimum Wiener ind_ex when k(?)6. And it should be noted that the result about the maximum Wiener ind_ex has been published in MATCH Commun. Math. Comput. Chem. 64(3)(2010).At last, we will summarize the whole paper and propose several problems that need thinking in the future.