On the Sum of Topological Index and Its Reciprocal Index of Trees with Fixed Matching Number

The sum of the degree-based topological index and its reciprocal index of a graph G is defined as T If (G) + RT If (G) = ∑︂ uv∈E(G) f(d(u), d(v)) + ∑︂ uv∈E(G) 1 f(d(u), d(v)), where f(x, y) = f(y, x) is a symmetric real-valued function. In this paper, for the first Zagreb index (f(x, y) = x + y), the forgotten index (f(x, y) = x 2 +y 2), and the Sombor index f(x, y) = √︁x2 + y2), we establish the sharp upper bounds of T If (G) +RT If (G) for trees with fixed order n and matching number m, and characterize the corresponding extremal graphs. These results extend the results on trees obtained by Gao [W. Gao, MATCH Commun. Math. Comput. Chem. 93 (2025) 535-547].