On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs
Let G be a graph of order n. Also let λ1≥λ2≥⋯≥λn be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k(≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with...
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| Veröffentlicht in: | Linear algebra and its applications 2019-05, Vol.569, p.175-194 |
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| creator | Das, Kinkar Chandra Mojallal, Seyed Ahmad Sun, Shaowei |
| description | Let G be a graph of order n. Also let λ1≥λ2≥⋯≥λn be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k(≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with equality holding if and only if G≅nK1 or G≅pK‾t(n=pt,p≥2) with k=1 (where pK‾t is the complete p-partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows:
Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=2n2(4d−n)2, where d>n4.
Finally we give a partial solution to this problem. |
| doi_str_mv | 10.1016/j.laa.2019.01.016 |
| format | Article |
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Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=2n2(4d−n)2, where d>n4.
Finally we give a partial solution to this problem.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2019.01.016</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Apexes ; Codes ; Eigenvalue sum ; Eigenvalues ; Graph ; Graph energy ; Graph theory ; Graphs ; Inertia ; Linear algebra ; Rank ; Upper bounds</subject><ispartof>Linear algebra and its applications, 2019-05, Vol.569, p.175-194</ispartof><rights>2019 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. May 15, 2019</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-84340f7bf5285da4693d3115803e63f6c8b94295577940cb0988c5cdd6a127a03</citedby><cites>FETCH-LOGICAL-c368t-84340f7bf5285da4693d3115803e63f6c8b94295577940cb0988c5cdd6a127a03</cites><orcidid>0000-0003-2576-160X</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.laa.2019.01.016$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Das, Kinkar Chandra</creatorcontrib><creatorcontrib>Mojallal, Seyed Ahmad</creatorcontrib><creatorcontrib>Sun, Shaowei</creatorcontrib><title>On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs</title><title>Linear algebra and its applications</title><description>Let G be a graph of order n. Also let λ1≥λ2≥⋯≥λn be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k(≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with equality holding if and only if G≅nK1 or G≅pK‾t(n=pt,p≥2) with k=1 (where pK‾t is the complete p-partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows:
Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=2n2(4d−n)2, where d>n4.
Finally we give a partial solution to this problem.</description><subject>Apexes</subject><subject>Codes</subject><subject>Eigenvalue sum</subject><subject>Eigenvalues</subject><subject>Graph</subject><subject>Graph energy</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Inertia</subject><subject>Linear algebra</subject><subject>Rank</subject><subject>Upper bounds</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp9UE1LxDAQDaLguvoDvAU8t06SJk3wJItfsLAXBW8hTdNua7etSXdx_71Zd8_CgxmY997MPIRuCaQEiLhv086YlAJRKZAIcYZmROYsIZKLczQDoFnCcsUv0VUILQBkOdAZ-lz1eFo7HLYbPFR_7RfujK9dmLBratfvTLd14TCsvRnXAZu-xBvz02xMh13vfL0_DItmNH5qJneiXaOLynTB3ZzqHH08P70vXpPl6uVt8bhMLBNySmTGMqjyouJU8tJkQrGSEcIlMCdYJawsVEYV53muMrAFKCktt2UpDKG5ATZHd0ff0Q_f8dBJt8PW93GlppQwEn2JiixyZFk_hOBdpUcfH_B7TUAfAtStjgHqQ4AaSISImoejxsXzd43zOtjG9daVjXd20uXQ_KP-BdPZdwE</recordid><startdate>20190515</startdate><enddate>20190515</enddate><creator>Das, Kinkar Chandra</creator><creator>Mojallal, Seyed Ahmad</creator><creator>Sun, Shaowei</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><orcidid>https://orcid.org/0000-0003-2576-160X</orcidid></search><sort><creationdate>20190515</creationdate><title>On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs</title><author>Das, Kinkar Chandra ; Mojallal, Seyed Ahmad ; Sun, Shaowei</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-84340f7bf5285da4693d3115803e63f6c8b94295577940cb0988c5cdd6a127a03</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Apexes</topic><topic>Codes</topic><topic>Eigenvalue sum</topic><topic>Eigenvalues</topic><topic>Graph</topic><topic>Graph energy</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Inertia</topic><topic>Linear algebra</topic><topic>Rank</topic><topic>Upper bounds</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Das, Kinkar Chandra</creatorcontrib><creatorcontrib>Mojallal, Seyed Ahmad</creatorcontrib><creatorcontrib>Sun, Shaowei</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>Das, Kinkar Chandra</au><au>Mojallal, Seyed Ahmad</au><au>Sun, Shaowei</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs</atitle><jtitle>Linear algebra and its applications</jtitle><date>2019-05-15</date><risdate>2019</risdate><volume>569</volume><spage>175</spage><epage>194</epage><pages>175-194</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let G be a graph of order n. Also let λ1≥λ2≥⋯≥λn be the eigenvalues of graph G. In this paper, we present the following upper bound on the sum of the k(≤n) largest eigenvalues of G in terms of the order n and negative inertia θ (the number of negative eigenvalues): ∑i=1kλi≤n2(θ+1)(θ+θ(kθ+k−1)) with equality holding if and only if G≅nK1 or G≅pK‾t(n=pt,p≥2) with k=1 (where pK‾t is the complete p-partite graph of p t vertices with all partition sets having size t). Moreover, we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r. We also propose an open problem as follows:
Characterize all connected d-regular bipartite graphs of order n with 5 distinct eigenvalues and rank r=2n2(4d−n)2, where d>n4.
Finally we give a partial solution to this problem.</abstract><cop>Amsterdam</cop><pub>Elsevier Inc</pub><doi>10.1016/j.laa.2019.01.016</doi><tpages>20</tpages><orcidid>https://orcid.org/0000-0003-2576-160X</orcidid><oa>free_for_read</oa></addata></record> |
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| subjects | Apexes Codes Eigenvalue sum Eigenvalues Graph Graph energy Graph theory Graphs Inertia Linear algebra Rank Upper bounds |
| title | On the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs |
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