Document Zbl 1246.05099

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W.</a>; <a href="/authors/sander.torsten" title="Author Profile">Sander, T.</a></div> <h2 class="title"> <strong>The maximal energy of classes of integral circulant graphs.</strong> <i>(English)</i> <a class="label nowrap" href="/1246.05099">Zbl 1246.05099</a> </h2> <div class="source"> <a href="/serials/68" title="Journal Profile">Discrete Appl. Math.</a> <a href="/?q=in%3A302816" title="Articles in this Issue">160, No. 13-14, 2015-2029 (2012)</a>. </div> <div class="abstract">Summary: The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs, which can be characterized by their vertex count $n$ and a set $D$ of divisors of $n$ in such a way that they have vertex set $\mathbb Z_{n}$ and edge set $\{\{a,b\}: a,b\in \mathbb Z_{n}, \text{gdc}(a-b)\in \mathcal D\}$.<br class="zbmathjax-paragraph"> For a fixed prime power $n=p^{s}$ and a fixed divisor set size $|\mathcal D|=r$, we analyse the maximal energy among all matching integral circulant graphs. Let $p^{a_{1}}<p^{a_{2}}< \cdots <p^{a_{r}}$ be the elements of $\mathcal D$. It turns out that the differences $d_{i}=a_{i+1} - a_{i}$ between the exponents of an energy maximal divisor set must satisfy certain balance conditions: (i) either all $d_{i}$ equal $q:=\frac {s-1}{r-1}$, or at most the two differences [$q$] and [$q+1$] may occur; (ii) there are rules governing the sequence $d_{1},\ldots ,d_{r - 1}$ of consecutive differences. For particular choices of s and r these conditions already guarantee maximal energy and its value can be computed explicitly.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A6064590">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C50" title="MSC2020">05C50</a> </td> <td class="space"> Graphs and linear algebra (matrices, eigenvalues, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C35" title="MSC2020">05C35</a> </td> <td class="space"> Extremal problems in graph theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A05C25" title="MSC2020">05C25</a> </td> <td class="space"> Graphs and abstract algebra (groups, rings, fields, etc.) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ACayley+graphs">Cayley graphs</a>; <a href="/?q=ut%3Aintegral+graphs">integral graphs</a>; <a href="/?q=ut%3Acirculant+graphs">circulant graphs</a>; <a href="/?q=ut%3Agcd+graphs">gcd graphs</a>; <a href="/?q=ut%3Agraph+energy">graph energy</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1246.05099" data-ciurl="/ci/06064590" data-biburl="/bibtex/06064590.bib" data-amsurl="/amsrefs/06064590.bib" data-xmlurl="/xml/06064590.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06064590.pdf" title="Zbl 1246.05099 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.dam.2012.04.017" aria-label="DOI for “The maximal energy of classes of integral circulant graphs”" title="10.1016/j.dam.2012.04.017">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1205.4603"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ahmadi, O.; Alon, N.; Blake, I. 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