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 application of these results, we obtain the bicyclic graph(s) with the smallest $f$-spectral radius for fixed order $n\ge 8$ by applying generalized Lu–Man method.