Speaker
Description
We study the $Kf- $ Šoltés problem, which is related to the resistance distance in a graph. While the original Šoltés problem deals with the identification of all graphs for which the removal of an arbitrary vertex preserves the Wiener index, the $Kf- $ Šoltés problem deals with graphs for which the removal of any vertex preserves the Kirchhoff index.
Currently, the only known solution to the $Kf- $Šoltés problem is the cycle $C_5$. We consider the relaxed version of the problem, which is called the $Kf_\beta\, - $ Šoltés problem: find graphs whose proportion of vertices that preserve the Kirchhoff index is equal to $\beta$. We show that for $0< \beta < 2/3$ the $Kf_{\beta}\, - $ Šoltés problem is rich with solutions. Namely, we construct an inifinite family of $Kf_{1/2}-$Šoltés graphs and build a family of graphs for which $\beta$ tends to $2/3$. We also study $Kf_{\beta}\,-$ Šoltés problem on unicyclic and bicyclic graphs.
