Graphical edge-weight-function indices of trees

Consider a tree graph $G$ with edge set $E(G)$. The notation $d_G(x)$ represents the degree of vertex $x$ in $G$. Let $\mathfrak{f}$ be a symmetric real-valued function defined on the Cartesian square of the set of all distinct elements of the degree sequence of $G$. A graphical edge-weight-function index for the graph $G$, denoted by ${\mathcal{I}}_{\mathfrak{f}}(G)$, is defined as ${\mathcal{I}}_{\mathfrak{f}}(G)=\sum _{st\in E(G)}\mathfrak{f}(d_G(s),d_G(t))$. This paper establishes the best possible bounds for ${\mathcal{I}}_{\mathfrak{f}}(G)$ in terms of the order of $G$ and parameter $\mathfrak{p}$, subject to specific conditions on $\mathfrak{f}$. Here, $\mathfrak{p}$ can be one of the following three graph parameters: (ⅰ) matching number, (ⅱ) the count of pendent vertices, and (ⅲ) maximum degree. We also characterize all tree graphs that achieve these bounds. The constraints considered for $\mathfrak{f}$ are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.