General Randić indices of a graph and its line graph

Abstract

For a real number α\alpha , the general Randić index of a graph GG, denoted by Rα(G){R}_{\alpha }\left(G), is defined as the sum of (d(u)d(v))α{\left(d\left(u)d\left(v))}^{\alpha } for all edges uvuv of GG, where d(u)d\left(u) denotes the degree of a vertex uu in GG. In particular, R−12(G){R}_{-\tfrac{1}{2}}\left(G) is the ordinary Randić index, and is simply denoted by R(G)R\left(G). Let α\alpha be a real number. In this article, we show that (1)if α≥0\alpha \ge 0, Rα(L(G))≥2Rα(G){R}_{\alpha }\left(L\left(G))\ge 2{R}_{\alpha }\left(G) for any graph GG with δ(G)≥3\delta \left(G)\ge 3;(2)if α≥0\alpha \ge 0, Rα(L(G))≥Rα(G){R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any connected graph GG which is not isomorphic to Pn{P}_{n};(3)if α<0\alpha \lt 0, Rα(L(G))≥Rα(G){R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any kk-regular graph GG with k≥2−2α+1k\ge {2}^{-2\alpha }+1;(4)R(L(S(G)))≥R(S(G))R\left(L\left(S\left(G)))\ge R\left(S\left(G)) for any graph GG with δ(G)≥3\delta \left(G)\ge 3, where S(G)S\left(G) is the graph obtained from GG by inserting exactly one vertex into each edge.

Keywords

• general randić index
• line graph
• randić index
• 05c09
• 05c76