## Abstract

For given graphs G and H, the graph G is H-saturated if G does not contain H as a subgraph but for any e∈E(G‾), G+e contains H. In this note, we prove that if G is an n-vertex K_{r+1}-saturated graph such that for each vertex v∈V(G), ∑w∈N(v)d_{G}(w)≥(r−2)d(v)+(r−1)(n−r+1), then ρ(G)≥ρ(S_{n,r}), where S_{n,r} is the graph obtained from a copy of K_{r−1} with vertex set S by adding n−r+1 vertices, each of which has neighborhood S. This provides a sharp lower bound for the spectral radius in an n-vertex K_{r+1}-saturated graph for r=2,3, verifying a special case of a conjecture by Kim, Kim, Kostochka and O.

Original language | English (US) |
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Article number | 113231 |

Journal | Discrete Mathematics |

Volume | 346 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2023 |

## Keywords

- Complete graphs
- Saturated graphs
- Spectral radius

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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