{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T17:35:13Z","timestamp":1776965713988,"version":"3.51.4"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>The inducibility of a graph $H$ is about the maximum number of induced copies of $H$ in a graph on $n$ vertices. We consider its edge version, that is, the maximum number of induced copies of $H$ in a graph with $m$ edges. Let $c(G,H)$ be the number of induced copies of $H$ in $G$ and $\\rho(H,m) = \\max \\{c(G,H) \\mid |E(G)| = m\\}$. For any graph $H$, we prove that $\\rho(H,m) = \\Theta(m^{\\alpha_f(H)})$ where $\\alpha_f(H)$ is the fractional independence number of $H$. Therefore, we now focus on the constant factor in front of $m^{\\alpha_f(H)}$. In this paper, we give some results of $\\rho(H,m)$ when $H$ is a cycle or path. We conjecture that for any cycle $C_k$ with $k \\ge 5$, $\\rho(C_k,m)= (1+o(1))\\left( m\/k\\right)^{k\/2}$ and the bound is achieved by the blow up of $C_k$. For even cycles, we establish an upper bound with an extra constant factor. For odd cycles, we can only establish an upper bound with an extra factor depending on $k$. We prove that $\\rho(P_{2l},m) \\le \\frac{m^l}{2(l-1)^{l-1}}$ and $\\rho(P_{2l+1},m) \\le \\frac{m^{l+1}}{4l^l}$, where $l \\ge 2$. We also conjecture the asymptotic value of $\\rho(P_k, m)$. The entropy method is mainly used to prove our results.<\/jats:p>","DOI":"10.37236\/14422","type":"journal-article","created":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T16:38:38Z","timestamp":1776962318000},"source":"Crossref","is-referenced-by-count":0,"title":["Edge Version of the Inducibility via the Entropy Method"],"prefix":"10.37236","volume":"33","author":[{"given":"Yichen","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiamiao","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mei","family":"Lu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2026,4,24]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p17\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T16:38:38Z","timestamp":1776962318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i2p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,24]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,4,14]]}},"URL":"https:\/\/doi.org\/10.37236\/14422","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,24]]},"article-number":"P2.17"}}