Spectral extremal graphs for intersecting cliques

The (k,r)-fan is the graph consisting of k copies of the complete graph Kr which intersect in a single vertex, and is denoted by Fk,r. Erdős et al. (1995) [14] determined the maximum number of edges in an n-vertex graph that does not contain Fk,3 as a subgraph. Furthermore, Chen et al. (2003) [5] pr...

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Published inLinear algebra and its applications Vol. 644; pp. 234 - 258
Main Authors Desai, Dheer Noal, Kang, Liying, Li, Yongtao, Ni, Zhenyu, Tait, Michael, Wang, Jing
Format Journal Article
LanguageEnglish
Published Amsterdam Elsevier Inc 01.07.2022
American Elsevier Company, Inc
Subjects
Extremal graph
Graph theory
Graphs
Intersecting cliques
Linear algebra
Spectral radius
Stability method
Intersecting cliques
05C50
Spectral radius
Extremal graph
Stability method
15A69
05C35
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Summary:The (k,r)-fan is the graph consisting of k copies of the complete graph Kr which intersect in a single vertex, and is denoted by Fk,r. Erdős et al. (1995) [14] determined the maximum number of edges in an n-vertex graph that does not contain Fk,3 as a subgraph. Furthermore, Chen et al. (2003) [5] proved the analogous result on Fk,r for the general case r≥3. In this paper, we show that for sufficiently large n, the graphs of order n that contain no copy of Fk,r and attain the maximum spectral radius are also edge-extremal. That is, such graphs must have ex(n,Fk,r) edges.
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content type line 14
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2022.03.015