{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:35Z","timestamp":1753893815155,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>For every graph $G,$ let $$ \\Delta_{r}\\left(G\\right)  =\\max\\left\\{  \\sum_{u\\in R}d\\left(  u\\right) :R\\hbox{ is an }r\\hbox{-clique of }G\\right\\} $$ and let $\\Delta_{r}\\left(  n,m\\right)  $ be the minimum of $\\Delta_{r}\\left( G\\right)$ taken over all graphs of order $n$ and size $m$. Write $t_{r}\\left(  n\\right)  $ for the size of the $r$-chromatic Tur\u00e1n graph of order $n$. Improving earlier results of Edwards and Faudree, we show that for every $r\\geq2,$ if $m\\geq t_{r}\\left(  n\\right)$, then $$ \\Delta_{r}\\left(  n,m\\right)  \\geq\\frac{2rm}{n},\\qquad(1) $$ as conjectured by Bollob\u00e1s and Erd\u0151s. It is known that inequality (1) fails for $m &lt; t_{r}\\left(n\\right)$.  However, we show that for every $\\varepsilon&gt;0,$ there is $\\delta&gt;0$ such that if $m&gt;t_{r}\\left(  n\\right)  -\\delta n^{2}$ then $$ \\Delta_{r}\\left(  n,m\\right)  \\geq\\left(  1-\\varepsilon\\right)  \\frac{2rm}{n}. $$<\/jats:p>","DOI":"10.37236\/1988","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:08:57Z","timestamp":1578719337000},"source":"Crossref","is-referenced-by-count":1,"title":["The Sum of Degrees in Cliques"],"prefix":"10.37236","volume":"12","author":[{"given":"B\u00e9la","family":"Bollob\u00e1s","sequence":"first","affiliation":[]},{"given":"Vladimir","family":"Nikiforov","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,11,7]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1n21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1n21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:40:09Z","timestamp":1579322409000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1n21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11,7]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1988","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,11,7]]},"article-number":"N21"}}