Remarks on the spectral radius of $K_{r+1}$-saturated graphs

Abstract

Write $\rho \left(G\right)$ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_r\vee {\overline{K}}_{n-r}.$ Let $n>r\ge 2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $\rho \left(G\right)$ for $r=2$, and found an asymptotically tight bound on $\rho \left(G\right)$ for $r\ge 3.$ They also conjectured that \[ ρ\left( G\right) >ρ\left( S_{n,r-1}\right) , \] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.

arXiv commentProposition 6 is wrong. There are counterexamples to the statement