2-Connected graphs with minimum general sum-connectivity index
The general sum-connectivity index of a graph G G is ? ? ( G ) = ? u v ? E ( G ) ( d ( u ) + d ( v ) ) ? , where d(u) d ( u ) denotes the degree of vertex u?V(G) u ? V ( G ) , and ? ? is a real number. In this paper, we show that in the class of graphs G G of order n?3 n ? 3 and minimum degree ?(G)?2 ? ( G ) ? 2 , the unique graph G G having minimum ??(G) ? ? ( G ) is K 2 + K n ? 2  if ?1??<?0??0.867 ? 1 ? ? < ? 0 ? ? 0.867 . Similarly, if we impose the supplementary condition for G G to be triangle-free, the extremal graph is K2,n?2 K 2 , n ? 2 for n?4 n ? 4 and ?1??<?0??0.817 ? 1 ? ? < ? 0 ? ? 0.817 . Since both extremal graphs are 2-connected, it follows that the properties are also true in the subclass of 2-connected graphs.
2-connected graph general sum-connectivity index minimum degree harmonic index jensens inequality concave function