{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T14:56:48Z","timestamp":1773068208995,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The $p$-spectral radius of a graph $G\\ $of order $n$ is defined for any real number $p\\geq1$ as$$\\lambda^{(p)}(G) =\\max\\{ 2\\sum_{\\{i,j\\}\\in E(G)} x_ix_j:x_1,\\ldots,x_n\\in\\mathbb{R}\\text{ and }\\vert x_{1}\\vert ^{p}+\\cdots+\\vert x_n\\vert^{p}=1\\} .$$The most remarkable feature of $\\lambda^{(p)}$ is that it\u00a0seamlessly joins several other graph parameters, e.g., $\\lambda^{(1)}$ is the Lagrangian, $\\lambda^{(2)\u00a0 }$ is the spectral\u00a0radius and $\\lambda^{(\\infty)\u00a0 }\/2$ is the number of edges. This\u00a0paper presents solutions to some extremal problems about $\\lambda^{(p)}$, which are common generalizations of corresponding edge and\u00a0spectral extremal problems.Let $T_{r}\\left(\u00a0 n\\right)\u00a0 $ be the $r$-partite Tur\u00e1n\u00a0graph of order $n$.\u00a0Two of the main results in the paper are:(I) Let $r\\geq2$ and $p&gt;1.$ If $G$ is a $K_{r+1}$-free graph of order $n$,\u00a0then$$\\lambda^{(p)}(G)\u00a0 &lt;\\lambda^{(p)}(T_{r}(n)),$$ unless $G=T_{r}(n)$.(II) Let $r\\geq2$ and $p&gt;1.$ If $G\\ $is a graph of order $n,$ with$$\\lambda^{(p)}(G)&gt;\\lambda^{(p)}(\u00a0 T_{r}(n))\u00a0 ,$$then $G$ has an edge contained in at least $cn^{r-1}$ cliques of order $r+1$,\u00a0where $c$ is a positive number depending only on $p$ and $r.$<\/jats:p>","DOI":"10.37236\/4113","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T00:56:38Z","timestamp":1578704198000},"source":"Crossref","is-referenced-by-count":4,"title":["Extremal Problems for the $p$-Spectral Radius of Graphs"],"prefix":"10.37236","volume":"21","author":[{"given":"Liying","family":"Kang","sequence":"first","affiliation":[]},{"given":"Vladimir","family":"Nikiforov","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2014,8,6]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p21\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v21i3p21\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T10:49:45Z","timestamp":1579258185000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v21i3p21"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,8,6]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2014,7,3]]}},"URL":"https:\/\/doi.org\/10.37236\/4113","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,8,6]]},"article-number":"P3.21"}}