General Randić matrix and general Randić incidence matrix

Let G be a connected graph with vertex set V(G) = {v₁,⋯, v_n} and edge set E(G) = {e₁,⋯, e_m}. Let d_i be the degree of the vertex ν_i. The general Randić matrix R_α = ((R_α)_ij)n×n of G is defined by (R_α)_ij = (d_id_j)^α if vertices ν_i and ν_j are adjacent in G and 0 otherwise. The Randić signless Laplacian matrix Q_α = D^2α+1 + R_α, where α is a nonzero real number and D is the degree diagonal matrix of G. The general Randić energy RE_α is the sum of absolute values of the eigenvalues of R_α. The general Randić incidence matrix B_Rα = ((B_Rα)_ij)n×m of a graph G is defined by (B_Rα)_ij = d^α_i if ν_i is incident to e_j and 0 otherwise. Naturally, the general Randić incidence energy BE_α is the sum of the singular values of B_Rα. In this paper, we investigate the connected graphs with s distinct R_α-eigenvalues, where 2 ≤ s ≤ n. Moreover, we establish the relation between the Randić signless Laplacian eigenvalues of G and the general Randić energy of its subdivided graph S (G). Also we give lower and upper bounds on the general Randić incidence energy. Finally, the general Randić incidence energy of a graph and that of its subgraphs are compared.