Graphical edge-weight-function indices of trees

Consider a tree graph G with edge set $ E(G) $. The notation dG(x) represents the degree of vertex x in G. Let 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 If(G), is defined as If(G) = ∑st ∈ E(G) f(dG(s), dG(t)). This paper establishes the best possible bounds for If(G) in terms of the order of G and parameter p, subject to specific conditions on f. Here, 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 f are satisfied by several well-known indices. We specifically illustrate our findings by applying them to the recently introduced Euler-Sombor index.