On the general sum-connectivity index of connected graphs with given order and girth

Abstract

In this paper, we show that in the class
of connected graphs $G$ of order $n\geq 3$ having girth at least equal to $k$, $3\leq k\leq n$, the unique graph $G$ having minimum general sum-connectivity index $\chi _{\alpha }(G)$ consists of $C_{k}$ and $n-k$ pendant vertices adjacent to a unique vertex of $C_{k}$, if $-1\leq \alpha <0$.
This property does not hold for zeroth-order general Randi\' c index $^{0}R_{\alpha}(G)$.