Another look at the degree-Kirchhoff index

Let G be an arbitrary graph with vertex set {1,2, ...,N} and degrees d _i ≤ D, for fixed D and all i, then for the index R′(G) = Σ_i<jd_jd_jR_ij we show that R′(G) ≥ 2 |E| (N - 2 + 1/D + 1). We also show that the minimum of R′(G) over all N-vertex graphs is attained for the star graph and its value is 2N² - 5N + 3.