Upper and Lower Bounds for the Mixed Degree-Kirchhoff Index

We introduce the mixed degree-Kirchhoff index, a new molecular descriptor defined by (Formula presented) where di is the degree of the vertex i and Rij is the effective resistance between vertices i and j. We give general upper and lower bounds for Ȓ(G) and show that, unlike other related descriptors, it attains its largest asymptotic value (order n4), among barbell graphs, for the highly asymmetric lollipop graph. We also give more refined lower (order n2) and upper (order n3) bounds for c-cyclic graphs in the cases 0 ≤ c ≤ 6. For this latter purpose we use a close relationship between our new mixed degree-Kirchhoff index and the inverse degree, prior bounds we found for the inverse degree of c-cyclic graphs, and suitable expressions for the largest and smallest effective resistances of c-cyclic graphs.