Abstract
The bond incident degree (BID) indices \( \mathcal{T}_{f}(G) \) of a connected graph \( G \) with edge-weight function \( f(x,y) \) are defined as \[ \mathcal{T}_{f}(G)=\sum_{v_{i}v_{j}\in E(G)}f(d(v_{i}),d(v_{j})), \] where \( f(x,y)>0 \) is a symmetric
real function with \( x\geq 1 \) and \( y\geq 1 \) and \( d(u) \) is the degree of vertex \( u \) in \( G \).
In this paper, we prove that extremal tree of order \( n \) with given independence number \( s\,\, (n/2\leq s\leq n-1) \) having maximum bond incident degree indices \( \mathcal{T}_{f} \) is the spur graph \( S_{n,s} \) if edge-weight
symmetric function \( f(x,y) \) satisfies three conditions: \( f(x,y) \) is strictly increasing on \( x \) (or \( y \)); \( f(x,y)-f(x,y-1) \) is increasing on \( x \) (or \( y \)); \( \varphi(x+1,y-1)\geq \varphi(x,y) \) for every \(
x, y\geq 2 \), where \( \varphi(x,y)=f(x,y)-f(x-1,y) \).
