Upper and Lower Bounds for the Mixed Degree-Kirchhoff Index

We introduce the mixed degree-Kirchhoff index, a new molecular descriptor defined by $$\widehat{R}(G)=\sum_{i<j}\left( {\frac{d_{i}}{d_{j}}}+{\frac{d_{j}}{d_{i}}}\right)R_{ij},$$ where $d_i$ is the degree of the vertex $i$ and $R_{ij}$ is the effective resistance between vertices $i$ and $j$. We give general upper and lower bounds for $\widehat{R}(G)$ and show that, unlike other related descriptors, it attains its largest asymptotic value (order $n^4$), among barbell graphs, for the highly asymmetric lollipop graph. We also give more refined lower (order $n^2$) and upper (order $n^3$) bounds for $c$-cyclic graphs in the cases $0\le c \le 6$. 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.