Conjectured bounds for the sum of squares of positive eigenvalues of a graph

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• 作者：Clive Elphick ^{clive.elphick@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Miriam Farber^a ; ^{mfarber@mit.edu" class="auth_mail" title="E-mail the corresponding author} ; Felix Goldberg^b ; ^{felix.goldberg@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Pawel Wocjan^c ; ^{wocjan@eecs.ucf.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C50 ; Adjacency matrix ; Inertia of a graph ; Hyper-energetic graphs
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 September 2016
• 年：2016
• 卷：339
• 期：9
• 页码：2215-2223
• 全文大小：374 K

文摘

A well known upper bound for the spectral radius of a graph, due to Hong, is that class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300024&_mathId=si6.gif&_user=111111111&_pii=S0012365X16300024&_rdoc=1&_issn=0012365X&md5=417956ccb4f74f06bc8b423d44ea1288">class="imgLazyJSB inlineImage" height="19" width="118" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0012365X16300024-si6.gif">class="mathContainer hidden">class="mathCode">${\mu }_1^2\le 2m-n+1$ if class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300024&_mathId=si7.gif&_user=111111111&_pii=S0012365X16300024&_rdoc=1&_issn=0012365X&md5=055baf3378ec278a16f98876a2a5d87d" title="Click to view the MathML source">δ≥1class="mathContainer hidden">class="mathCode">$\delta \ge 1$. It is conjectured that for connected graphs class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300024&_mathId=si8.gif&_user=111111111&_pii=S0012365X16300024&_rdoc=1&_issn=0012365X&md5=f35a6d5231b1f920260621756a786067" title="Click to view the MathML source">n−1≤s⁺≤2m−n+1class="mathContainer hidden">class="mathCode">$n-1\le s^{+}\le 2m-n+1$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300024&_mathId=si9.gif&_user=111111111&_pii=S0012365X16300024&_rdoc=1&_issn=0012365X&md5=072f53a451c101dd17fffa1e9a816501" title="Click to view the MathML source">s⁺class="mathContainer hidden">class="mathCode">$s^{+}$ denotes the sum of the squares of the positive eigenvalues. The conjecture is proved for various classes of graphs, including bipartite, regular, complete class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0012365X16300024&_mathId=si10.gif&_user=111111111&_pii=S0012365X16300024&_rdoc=1&_issn=0012365X&md5=ab250cd39257882601d4428a11b62657" title="Click to view the MathML source">qclass="mathContainer hidden">class="mathCode">$q$-partite, hyper-energetic, and barbell graphs. Various searches have found no counter-examples. The paper concludes with a brief discussion of the apparent difficulties of proving the conjecture in general.