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(dᵢ,dⱼ), where f is a real symmetric function and dᵢ,dⱼ 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≥8 by applying generalized Lu-Man method.