Abstract
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.
Lingua originale | English |
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pagine (da-a) | 2351-2358 |
Numero di pagine | 8 |
Rivista | Filomat |
DOI | |
Stato di pubblicazione | Pubblicato - 2016 |
Keywords
- Inverse degree
- Majorization