Minimum bond incident degree indices of unicyclic graphs with fixed girth and maximum degree

For a graph $G$, its bond incident degree index is given by

$$BID=\sum _{uv\in E(G)}f(d(u),d(v)),$$

in which $f(x,y)=f(y,x)$ is a real-valued symmetric function. In this paper, we establish three sufficient conditions respectively for the minimum bond incident degree ( $BID$) index of unicyclic graphs with given girth and those with given maximum degree. As applications, we verify that seven topological indices, including the recently proposed hyperbolic Sombor index and diminished Sombor index satisfy these sufficient conditions. This work provides a theoretical foundation for further investigations into the extremal values of topological indices.