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On the bounds of the forgotten topological index

In this paper, we obtain, analyze, and compare various lower bounds for the forgotten topological index involving the number of vertices, edges, and maximum and minimum vertex degree. Then we give Nordhaus–Gaddumtype inequalities for the forgotten topological index and coindex. Finally, we correct the number of extremal chemical trees on 15 vertices. | Turk J Math (2017) 41: 1687 – 1702 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1610-127 Research Article On the bounds of the forgotten topological index 1 Suresh ELUMALAI1,∗, Toufik MANSOUR2 , Mohammad Ali ROSTAMI3 Department of Mathematics, Velammal Engineering College, Surapet, Chennai, Tamil Nadu, India 2 Department of Mathematics, University of Haifa, Haifa, Israel 3 Institute for Computer Science, Friedrich Schiller University, Jena, Germany Received: 31.10.2016 • • Accepted/Published Online: 19.02.2017 Final Version: 23.11.2017 Abstract: The forgotten topological index is defined as the sum of cubes of the degrees of the vertices of the molecular graph G. In this paper, we obtain, analyze, and compare various lower bounds for the forgotten topological index involving the number of vertices, edges, and maximum and minimum vertex degree. Then we give Nordhaus–Gaddumtype inequalities for the forgotten topological index and coindex. Finally, we correct the number of extremal chemical trees on 15 vertices. Key words: First Zagreb index, second Zagreb index, forgotten topological index 1. Introduction Throughout this paper, we consider G to be a simple connected graph with |V (G)| = n vertices and |E(G)| = m edges. The degree of a vertex vi (1 ≤ i ≤ n) is denoted by d(vi ) such that d(v1 ) ≥ d(v2 ) ≥ · · · ≥ d(vn ) . In particular, ∆, ∆2 , and δ are called the first, second maximum, and minimum degrees of G , respectively. Let G denote the complement graph of G with the same vertex set V (G) in which two vertices u and v are adjacent if and only if they are not adjacent in G . The line graph L(G) is obtained from G in which V (L(G)) = E(G), where two vertices of L(G) are adjacent if and only if they are adjacent edges of G . In 1972, Gutman and Trinajsti´c introduced the classical Zagreb indices in [13] and they are among the oldest and most used molecular structure-descriptors. The first Zagreb