The Extremal Values of Topological Index of Caterpillars with Given Degree Sequence
|School||Shanghai Jiaotong University|
|Keywords||topological index caterpillar Randi? index Wiener index|
Randi（?） index and Wiener index are two important molecular topological indices in chemical graph theory. Moreover, in the general research about graph theory, they also have important theoretical significances. So in this paper we will study the two indices. In the paper, we will discuss some properties of the extremal caterpillars which respectively attain the extremal values of Randi（?） index and Wiener index among all caterpillars with given degree sequence. And according to the properties, the structure of the extremal caterpillars will become clear.Firstly, we will use the methods in algebra and graph theory to characterize the extremal caterpillar with the maximum/minimum Randi（?） index among all caterpillars with given degree sequenceπ=（ d1 , d 2, ..., dn）, where d1 （?） d 2 ?...（?） d k （?） 2（?） dk+1=...=dn=1 Moreover, subsequently we will learn that when k is fixed the extremal caterpillar with the maximum/minimum Randi（?） index is unique. In addition, we will point out the condition“d1 （?） d 2 ?...?dk”is indispensable.Secondly, we will consider the Wiener index of the caterpillars with given degree sequenceπ=（ d1 , d 2, ..., dn）, where d1 （?） d 2 ?...（?） d k （?） 2（?） dk+1=...=dn=1 and characterize the extremal caterpillars which respectively attain the maximum and minimum Wiener index when k（?）6. And it should be noted that the result about the maximum Wiener index 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.