On the sum-connectivity index of cacti

Abstract

For a simple connected graph G = ( V , E ) , X ( G ) = ∑ u v ∈ E 1 d u + d v is its sum-connectivity index, where d u denotes the degree of a vertex u . A connected graph G is a cactus if any two of its cycles have at most one common vertex. Let G ( n , r ) be the set of cacti of order n and with r cycles, ζ ( 2 n , r ) the set of cacti of order 2 n with a perfect matching and r cycles. In this paper, we give the sharp lower bounds of the sum-connectivity index of cacti among G ( n , r ) and ζ ( 2 n , r ) respectively: (1) if G ∈ G ( n , r ) , n ≥ 5 , then X ( G ) ≥ 2 r n + 1 + n − 2 r − 1 n + r 2 ; (2) if G ∈ ζ ( 2 n , r ) , n ≥ 4 , then X ( G ) ≥ n + r − 1 n + r + 2 + 1 n + r + 1 + n − r − 1 3 + r 2 , and characterize the corresponding extremal cacti.