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The spectral radius of graphs without long cycles
Let Cℓ be a cycle of length ℓ and Sn,k=Kk∨Kn−k‾, the join graph of a complete graph of order k and an empty graph on n−k vertices, and Sn,k+ be the graph obtained from Sn,k by adding an edge in the independent set of Sn,k. Nikiforov conjectured that for a given integer k≥2, any graph G of sufficient...
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| Published in: | Linear algebra and its applications 2019-04, Vol.566, p.17-33 |
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| Language: | English |
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| container_end_page | 33 |
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| container_title | Linear algebra and its applications |
| container_volume | 566 |
| creator | Gao, Jun Hou, Xinmin |
| description | Let Cℓ be a cycle of length ℓ and Sn,k=Kk∨Kn−k‾, the join graph of a complete graph of order k and an empty graph on n−k vertices, and Sn,k+ be the graph obtained from Sn,k by adding an edge in the independent set of Sn,k. Nikiforov conjectured that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains C2k+1 or C2k+2 (or C2k+2), unless G=Sn,k (or G=Sn,k+). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains a cycle Cℓ with ℓ≥2k+1 (or Cℓ with ℓ≥2k+2), unless G=Sn,k (or G=Sn,k+). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2]. |
| doi_str_mv | 10.1016/j.laa.2018.12.023 |
| format | article |
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Nikiforov conjectured that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains C2k+1 or C2k+2 (or C2k+2), unless G=Sn,k (or G=Sn,k+). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains a cycle Cℓ with ℓ≥2k+1 (or Cℓ with ℓ≥2k+2), unless G=Sn,k (or G=Sn,k+). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2].</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2018.12.023</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Apexes ; Cycle ; Extremal problem ; Graph theory ; Linear algebra ; Spectra ; Spectral radius</subject><ispartof>Linear algebra and its applications, 2019-04, Vol.566, p.17-33</ispartof><rights>2018 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. 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Nikiforov conjectured that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains C2k+1 or C2k+2 (or C2k+2), unless G=Sn,k (or G=Sn,k+). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains a cycle Cℓ with ℓ≥2k+1 (or Cℓ with ℓ≥2k+2), unless G=Sn,k (or G=Sn,k+). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2].</description><subject>Apexes</subject><subject>Cycle</subject><subject>Extremal problem</subject><subject>Graph theory</subject><subject>Linear algebra</subject><subject>Spectra</subject><subject>Spectral radius</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9kMtOwzAQRS0EEqXwAewisU6YsRMnEStU8ZIqsSlry7EnraPQBDsB9e9xVdasZnPPvaPD2C1ChoDyvst6rTMOWGXIM-DijC2wKkWKVSHP2QKA56ko6-KSXYXQAUBeAl8w3OwoCSOZyes-8dq6OSRDm2y9Hnch-XHTbpinpB_228QcTE_hml20ug9083eX7OP5abN6TdfvL2-rx3VqhKymVDcNxIWm4RVSbmxtCGugVsi60qa0KFsrbV6UKGVuSZuiFrzRdd4Uoi0IxJLdnXpHP3zNFCbVDbPfx0nFsRYSOVR5TOEpZfwQgqdWjd59an9QCOpoRnUqmlFHMwq5imYi83BiKL7_7cirYBztDVnnowhlB_cP_QtoVWqg</recordid><startdate>20190401</startdate><enddate>20190401</enddate><creator>Gao, Jun</creator><creator>Hou, Xinmin</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20190401</creationdate><title>The spectral radius of graphs without long cycles</title><author>Gao, Jun ; Hou, Xinmin</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-abb0702bb281e4cd9ce190ef3698ac7d16fd6d4571664deac5932ba94b53f5e03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Apexes</topic><topic>Cycle</topic><topic>Extremal problem</topic><topic>Graph theory</topic><topic>Linear algebra</topic><topic>Spectra</topic><topic>Spectral radius</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Gao, Jun</creatorcontrib><creatorcontrib>Hou, Xinmin</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Gao, Jun</au><au>Hou, Xinmin</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The spectral radius of graphs without long cycles</atitle><jtitle>Linear algebra and its applications</jtitle><date>2019-04-01</date><risdate>2019</risdate><volume>566</volume><spage>17</spage><epage>33</epage><pages>17-33</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let Cℓ be a cycle of length ℓ and Sn,k=Kk∨Kn−k‾, the join graph of a complete graph of order k and an empty graph on n−k vertices, and Sn,k+ be the graph obtained from Sn,k by adding an edge in the independent set of Sn,k. Nikiforov conjectured that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains C2k+1 or C2k+2 (or C2k+2), unless G=Sn,k (or G=Sn,k+). In this paper, a weaker version of Nikiforov's conjecture is considered, we prove that for a given integer k≥2, any graph G of sufficiently large order n with spectral radius μ(G)≥μ(Sn,k) (or μ(G)≥μ(Sn,k+)) contains a cycle Cℓ with ℓ≥2k+1 (or Cℓ with ℓ≥2k+2), unless G=Sn,k (or G=Sn,k+). These results also imply a result given by Nikiforov in (2010) [3, Theorem 2].</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2018.12.023</doi><tpages>17</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Linear algebra and its applications, 2019-04, Vol.566, p.17-33 |
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| language | eng |
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| source | ScienceDirect Freedom Collection 2022-2024 |
| subjects | Apexes Cycle Extremal problem Graph theory Linear algebra Spectra Spectral radius |
| title | The spectral radius of graphs without long cycles |
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