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 in	Linear 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
Language	English
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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Abstract	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.
AbstractList	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. 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.
Author	Tait, Michael Ni, Zhenyu Desai, Dheer Noal Li, Yongtao Wang, Jing Kang, Liying
Author_xml	– sequence: 1 givenname: Dheer Noal orcidid: 0000-0002-2529-8937 surname: Desai fullname: Desai, Dheer Noal email: dheernsd@udel.edu organization: Department of Mathematical Sciences, University of Delaware, USA – sequence: 2 givenname: Liying orcidid: 0000-0002-7752-0392 surname: Kang fullname: Kang, Liying email: lykang@shu.edu.cn organization: Department of Mathematics, Shanghai University, Shanghai 200444, PR China – sequence: 3 givenname: Yongtao surname: Li fullname: Li, Yongtao email: ytli0921@hnu.edu.cn organization: School of Mathematics, Hunan University, Changsha, PR China – sequence: 4 givenname: Zhenyu surname: Ni fullname: Ni, Zhenyu email: 1051466287@qq.com organization: Department of Mathematics, Hainan University, Haikou 570228, PR China – sequence: 5 givenname: Michael orcidid: 0000-0003-3695-6883 surname: Tait fullname: Tait, Michael email: michael.tait@villanova.edu organization: Department of Mathematics and Statistics, Villanova University, USA – sequence: 6 givenname: Jing surname: Wang fullname: Wang, Jing email: wj517062214@163.com organization: Department of Mathematics, Shanghai University, Shanghai 200444, PR China
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Snippet	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]...
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SubjectTerms	Extremal graph Graph theory Graphs Intersecting cliques Linear algebra Spectral radius Stability method
Title	Spectral extremal graphs for intersecting cliques
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