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

The variable connectivity index, introduced by the chemist Milan Randic in the first quarter of 1990s, for a graph G is defined as Sigma(vw is an element of E(G))((d(v) + gamma)(d(w) + gamma))(-1/2), where gamma is a non-negative real number and d(w) is the degree of a vertex w in G. We call this index as the variable Randic index and denote it by R-v(gamma). In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum R-v(gamma) value among all unicyclic graphs of a fixed order n, for every n >= 4 and gamma >= 0.