weighted topological indices of graphs

The definition of the weighted topological index associated with a degree function $phi$ is $phi(g)=sum_{uvin e(g)}phi(d_{u},d_{v})$, where $d_{u}$ denotes the degree of node $u$ and $phi$ satisfies symmetric property $phi(d_{u},d_{v})=phi(d_{v},d_{u})$. in this paper, we characterized extremal graphs and presented several results concerning the function $phi(g)$ in terms of various graph invariants. additionally, we characterize the graphs that achieve these bounds and present multiple bounds for $phi(g)$ for the class of cozero divisor graphs defined on commutative rings.