Higher-order Randić index and isomorphism of double starlike trees

Abstract

<abstract><p>For an integer $ h\geq 0 $, the $ h $th order Randić index for a simple graph $ G $ is defined as $ R^{h}(G) = \sum_{\pi}\frac{1}{\sqrt{v_1(\pi)v_2(\pi)\cdots v_{h+1}(\pi)}} $, where $ \pi $ extends over all paths of length $ h $ in $ G $ and $ v_i(\pi) $ denotes the degree of the $ i $-th vertex of the path $ \pi $. In this paper, we showed that the $ h $th order Randić index $ R^{h}(T) $ of a double starlike tree $ T $ (a tree with two vertices of degrees $ m_1, m_2 &gt; 2 $) is completely determined by its branches of length $ \leq h $. As a consequence, we proved that the double starlike trees with equal $ h $-Randić index are isomorphic, except for some special values for $ m_1 $ and $ m_2 $.</p></abstract>