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تسجيل مستخدم جديد$K_{r+1}$-saturated graphs with small spectral radius
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For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e in E(overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $ntoinfty$.
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Write $rholeft( Gright) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}veeoverline{K}_{n-r}.$ Let $n>rgeq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the
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