Degree-Based Function Index of Graphs with Given Connectivity

‎We investigate the‎~ ‎index $I_{f} (G) = \sum_{vw \in E(G)} f(d_G (v),d_G (w))$ of a graph $G$‎, ‎where $f$ is a symmetric function of two variables satisfying certain conditions‎, ‎$E(G)$ is the edge set of $G$‎, ‎and $d_G (v)$ and $d_G (w)$ are the degrees of vertices $v$ and $w$ in $G$‎, ‎respectively‎. ‎Those conditions are satisfied by functions that can be used to define the general sum-connectivity index $\chi_{a}$‎, ‎general Randi\'{c} index $R_{a}$‎, ‎general reduced second Zagreb index $GRM_a$ for some $a \in \mathbb{R}$‎, ‎general Sombor index $SO_{a,b}$‎, ‎general augmented Zagreb index $AZI_{a,b}$ and by one other generalization $M_{a,b}$ for some $a‎, ‎b \in \mathbb{R}$‎. ‎The general augmented Zagreb index is a new index defined in this paper‎. ‎We obtain a sharp upper bound on $I_f$ for graphs with given order and connectivity‎, ‎and a sharp lower bound on $I_f$ for $2$-connected graphs with given order‎. ‎Our upper bound holds for $M_{a,b}$ and $SO_{a,b}$ where $a‎, ‎b \ge 1$; $\chi_a$ and $R_a$ where $a \ge 1$; and $GRM_{a}$ where $a >‎ -1$. ‎Our lower bound holds for $M_{a,b}$ where $a \ge 0$ and $b \ge‎ -‎a$; $SO_{a,b}$ where $a‎, ‎b \ge 0$ or $a‎, ‎b \le 0$; $AZI_{a,b}$ where $a \ge‎ -‎2$ and $b \ge 0$; $\chi_a$ and $R_a$ where $a \ge 0$; and $GRM_{a}$ where $a >‎ -‎2$.