A Unified Approach to Extremal Trees with Respect to Geometric–Arithmetic, Szeged and Edge Szeged Indices
نویسندگان
چکیده
The second and third geometric-arithmetic indices GA2(G) and GA3(G) of a graph G are defined, respectively, as ∑ uv∈E(G) √ nu(e,G)nv(e,G) 1 2 [nu(e,G)+nv(e,G)] and ∑ uv∈E(G) √ mu(e,G)mv(e,G) 1 2 [mu(e,G)+mv(e,G)] , where e = uv is one edge in G, nu(e,G) denotes the number of vertices in G lying closer to u than to v andmu(e,G) denotes the number of edges in G lying closer to u than to v. The Szeged and edge Szeged indices are defined, respectively, as Sz(G) = ∑ uv∈E(G) nu(e,G) · nv(e,G) and Sze(G) = ∑ uv∈E(G) mu(e,G) · mv(e,G). In this paper, we provide a unified approach to characterize the tree with the minimum and maximum GA2, GA3, Sz and Sze indices among the set of trees with given order and pendent vertices, respectively. As applications, we deduce a result of [2] concerning tree with the maximum GA2 index and a result of [3] concerning tree with the maximum GA3 index. ∗Supported by NSFC (No. 10871158) and Qing Lan Project of Jiangsu Province, P.R. China. †E-mail: [email protected] (H. Hua), [email protected] (S. Zhang) MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 65 (2011) 691-704 ISSN 0340 6253
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