General Randić indices of a graph and its line graph

Abstract

Abstract For a real number α \alpha , the general Randić index of a graph G G , denoted by R α ( 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 u v uv of G G , where d ( u ) d\left(u) denotes the degree of a vertex u u in G G . In particular, R − 1 2 ( 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 , R α ( L ( G ) ) ≥ 2 R α ( G ) {R}_{\alpha }\left(L\left(G))\ge 2{R}_{\alpha }\left(G) for any graph G G with δ ( G ) ≥ 3 \delta \left(G)\ge 3 ; (2) if α ≥ 0 \alpha \ge 0 , R α ( L ( G ) ) ≥ R α ( G ) {R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any connected graph G G which is not isomorphic to P n {P}_{n} ; (3) if α < 0 \alpha \lt 0 , R α ( L ( G ) ) ≥ R α ( G ) {R}_{\alpha }\left(L\left(G))\ge {R}_{\alpha }\left(G) for any k k -regular graph G G with k ≥ 2 − 2 α + 1 k\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 G G with δ ( G ) ≥ 3 \delta \left(G)\ge 3 , where S ( G ) S\left(G) is the graph obtained from G G by inserting exactly one vertex into each edge.