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The connectivity index, introduced by the chemist Milan Randić in 1975, is one of the topological indices with many applications. In the first quarter of 1990s, Randić proposed the variable connectivity index by extending the definition of the connectivity index. The variable connectivity index for graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a nonnegative real number, EG is the edge set of G, and dt denotes the degree of an arbitrary vertex t in G. Soon after the innovation of the variable connectivity index, its various chemical applications have been reported in different papers. However, to the best of the authors’ knowledge, mathematical properties of the variable connectivity index, for γ>0, have not yet been discussed explicitly in any paper. The main purpose of the present paper is to fill this gap by studying this topological index in a mathematical point of view. More precisely, in this paper, we prove that the star graph has the minimum variable connectivity index among all trees of a fixed order n, where n≥4.

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The total irregularity index of a graph [Formula: see text] is defined by Abdo et al. [H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014) 201–206] as [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In 2014, You et al. [L. H. You, J. S. Yang and Z. F. You, The maximal total irregularity of unicyclic graphs, Ars Comb. 114 (2014) 153–160.] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Unicyclic graphs) and Zhou et al. [L. H. You, J. S. Yang, Y. X. Zhu and Z. F. You, The maximal total irregularity of bicyclic graphs, J. Appl. Math. 2014 (2014) 785084, http://dx.doi.org/10.1155/2014/785084 ] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Bicyclic graphs). In this paper, we characterize the aforementioned graphs with an alternative but comparatively simple approach. Also, we characterized the graphs having maximum [Formula: see text] value among the classes [Formula: see text] (Tricyclic graphs), [Formula: see text] (Tetracyclic graphs), [Formula: see text] (Pentacyclic graphs) and [Formula: see text] (Hexacyclic graphs).