tailieunhanh - Correcting a paper on the Randic and geometric–arithmetic indices

In the paper, we first point out that Theorems 1, 2, and 4 are incorrect and in this short note we present the correct inequalities for Randic and GA indices. In the same paper, we provide the equality cases for Theorems 3, 5, and 6. | Turk J Math (2017) 41: 27 – 32 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Correcting a paper on the Randi´ c and geometric–arithmetic indices Toufik MANSOUR1 , Mohammad Ali ROSTAMI2 , Suresh ELUMALAI3,∗, Britto Antony XAVIER4 1 Department of Mathematics, University of Haifa, Haifa, Israel 2 Institute for Computer Science, Friedrich Schiller University Jena, Germany 3 Department of Mathematics, Velammal Engineering College, Surapet, Chennai, Tamil Nadu, India 4 Department of Mathematics, Sacred Heart College,Tirupattur, Tamil Nadu, India Received: • Accepted/Published Online: • Final Version: Abstract: The Randi´c index (R) and the geometric–arithmetic index (GA) are found to be useful tools in QSPR and QSAR studies. In the Journal of Inequalities and Applications 180, 1-7, Lokesha, Shwetha Shetty, Ranjini, Cangul, and Cevik gave ”New bounds for Randi´c and GA indices.” In the paper, we first point out that Theorems 1, 2, and 4 are incorrect and in this short note we present the correct inequalities for Randi´c and GA indices. In the same paper, we provide the equality cases for Theorems 3, 5, and 6. Key words: Randi´c index, geometric–arithmetic index, Zagreb index 1. Introduction Let G be a simple graph with n vertices and m edges. The degree of a vertex is denoted by d(vi ), for i = 1, 2, . . . , n such that d(v1 ) ≥ d(v2 ) ≥ · · · ≥ d(vn ) , with the maximum and the minimum vertex degree of G denoted by ∆ = ∆(G) and δ = δ(G), respectively. A vertex v ∈ V (G) is said to be pendant if its neighborhood contains exactly one vertex, . d(v) = 1. Moreover, p and δ1 = δ1 (G) denotes the number of pendant vertices and minimum nonpendant vertex degree in G , respectively. A graph G is called bidegreed if its vertex degree is either ∆ or δ with ∆ > δ ≥ 1 and Kr,n−r (1 ≤ r ≤ n − 1) denotes the bidegreed bipartite graph with r vertices of degree ∆