Abstract
In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph G, its vertex-degree-based topological indices of the form BID (G) = Sigma(uv epsilon E(G))beta(d(u), d(v)) are known as bond incident degree indices, where E (G) is the edge set of G, d(w) denotes degree of an arbitrary vertex w of G and beta is a real-valued-symmetric function. Those BID indices for which can be rewritten as a function of d(u) + d(v) - 2 (that is degree of the edge uv) are known as edge-degree-based BID indices. A connected graph G is said to be r-apex tree if r is the smallest nonnegative integer for which there is a subset R of V (G) such that vertical bar R vertical bar = r and G - R is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary BID index from the class of all r-apex trees of order n, where r and n are fixed integers satisfying the inequalities n - r >= 2 and r >= 1.