Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size

The $f$ adjacency matrix is a type of edge-weighted adjacency matrix, whose weight of an edge $ij$ is $f(d_i,d_j)$, where $f$ is a real symmetric function and $d_i,d_j$ are the degrees of vertex $i$ and vertex $j$. The $f$-spectral radius of a graph is the spectral radius of its $f$-adjacency matrix. In this paper, the effect of subdividing an edge on $f$-spectral radius is discussed. Some necessary conditions of the extremal graph with given order and size are derived. As an example, we obtain the bicyclic graph(s) with the smallest $f$-spectral radius for fixed order $n\ge 8$ by applying generalized Lu-Man method.