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

TL;DR

This paper proves a conjecture regarding the spectral radius of $K_{r+1}$-saturated graphs, showing it exceeds that of a specific join graph unless the graph is that join, extending prior results for all $r \

Contribution

The paper confirms a conjecture that the spectral radius of $K_{r+1}$-saturated graphs exceeds that of a particular join graph unless it is that graph, generalizing previous findings.

Findings

Proved the conjecture for all $r \\geq 2$.

Established the spectral radius bound for $K_{r+1}$-saturated graphs.

Extended prior results from specific cases to general $r$.

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 $n>r\geq2$ 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 $r\geq3.$ They also conjectured that \[ \rho\left( G\right) >\rho\left( S_{n,r-1}\right) , \] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.