Unicyclic graphs of given girth k≥4k≥4 having smallest general sum-connectivity index

The general sum-connectivity index of a graph GG is defined as χα(G)=∑uv∈E(G)(d(u)+d(v))αχα(G)=∑uv∈E(G)(d(u)+d(v))α, where d(u)d(u) denotes the degree of vertex uu in GG, and αα is a real number. In this paper, we characterize nn-vertex connected unicyclic graphs with girth k≥4k≥4, having the minimum, the second and the third minimum general sum-connectivity indices for n≥8n≥8 and −1≤α<0−1≤α<0.