Minimum Variable Connectivity Index of Trees of a Fixed Order

Abstract

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.

Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

“…by taking α as any real number and l as any nonnegative integer. We note that the graph invariant (3) (i) Remains well-defined if l is any real number greater than −1 (ii) Gives the reduced second Zagreb index [11] when one takes α � 1 and l � −1 (iii) Coincides with the variable connectivity index [26][27][28] if α � −1/2 and l is any nonnegative real number us, in what follows, we assume that (l, α) ∈ (A × R) ∪ (B × R + ) and call the graph invariant (3) as the Bollobás-Erdős-Sarkar index and denote it by BES (l,α) , where A is the set of all real numbers greater than −1, R is the set of all real numbers, R + is the set of all positive real numbers, and B � −1 { }. us, the Bollobás-Erdős-Sarkar index of a graph G is defined as…”

On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order

“…Details about the chemical applications of the variable Randić index can be found in [4,7,[9][10][11][12][13][14][15][16][17] and related references listed therein. In [18], a mathematical study of the variable Randić index was initiated and it was proved that the star graph has the minimum variable Randić index among all trees of a fixed order n, where n ≥ 4. It needs to be mentioned here that the variable Randić index seems to have more chemical applications than the several wellknown variable indices, see, for example, the variable indices considered in the papers [19][20][21][22][23][24][25].…”