Upper and Lower Bounds for the Mixed Degree-Kirchhoff Index

Monica Bianchi, Alessandra Cornaro, José Luis Palacios, Anna Torriero


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 an 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.

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