{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T18:04:23Z","timestamp":1758823463882,"version":"3.41.2"},"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>A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs.\u00a0In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph $G$ with diameter $k$ and vertex set $V$, the smallest nontrivial eigenvalue $\\lambda_1$ of the normalized Laplacian $\\mathcal L$ satisfies$$ \\lambda_1 \\leq 1-\\sigma \\big(1- \\frac c {k} \\big)$$\u00a0for some constant $c$\u00a0where $\\sigma = 2\\sum_v d_v \\sqrt{d_v-1}\/\\sum_v d_v^2 $ and $d_v$ denotes the degree of the vertex $v$.We consider weak Ramanujan graphs defined as graphs satisfying $ \\lambda_1 \\geq 1-\\sigma$.\u00a0We examine the vertex expansion and edge expansion of weak Ramanujan graphs and then use the expansion properties among other methods to derive the above Alon-Boppana bound.\u00a0A corrigendum was added on the 3rd of November 2017.<\/jats:p>","DOI":"10.37236\/5933","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:49:01Z","timestamp":1578689341000},"source":"Crossref","is-referenced-by-count":2,"title":["A Generalized Alon-Boppana Bound and Weak Ramanujan Graphs"],"prefix":"10.37236","volume":"23","author":[{"given":"Fan","family":"Chung","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,7,8]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p4\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p4\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:27:09Z","timestamp":1579238829000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i3p4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,7,8]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,7,8]]}},"URL":"https:\/\/doi.org\/10.37236\/5933","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,7,8]]},"article-number":"P3.4"}}