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

Ioan Tomescu (Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei, 14, Bucharest, Romania)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 1 (2016): Electronic Journal of Graph Theory and Applications

In this paper, we show that in the classof 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 $^{0}R_{\alpha}(G)$.