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

Abstract

The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

Minimum Variable Connectivity Index of Trees of a Fixed Order

“…Liu and Zhong [5] showed that the variable Randić index has more flexibility in characterizing polymers, which can lead to simpler correlations with better correlative accuracy. Details about the chemical applications of the variable Randić index can be found in [5,7,9,12,[15][16][17][18][19][20][21][22] and related references listed therein. It needs to be mentioned here that the variable Randić index seems to have more chemical applications than the several well-known variable indices, for example, the indices considered in [23][24][25][26][27][28][29][30][31][32][33].…”