{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:33Z","timestamp":1753893813801,"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 a given graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph, but for $e \\in E(\\overline{G})$, $G+e$ contains $H$ as a subgraph; the spectral saturation number of $H$, written $sat_{\\rho}(n,H)$, is the minimum value of $\\rho(G)$ in an $n$-vertex $H$-saturated graph $G$.\r\nFor a vertex $v \\in V(G)$, let $l_2(v)$ be the number of $2$-walks starting from $v$. In this paper, when $G$ is an $n$-vertex $tP_3$- or $K_5$-saturated connected graph, for each vertex $v \\in V(G)$, we prove the best lower bounds for $l_2(v)$ in terms of $n$ and $d(v)$, implying that $sat_{\\rho}(n,tP_3)=\\rho(F)$ and $sat_{\\rho}(n,K_5)=\\rho(S_{n,4})$, where $F$ is the $6$-vertex graph obtained from $K_3$ by attaching a pendant vertex to each vertex in $K_3$ and $S_{n,4}$ is the join of $K_3$ and $(n-3)K_1$.<\/jats:p>","DOI":"10.37236\/12492","type":"journal-article","created":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T11:22:41Z","timestamp":1740655361000},"source":"Crossref","is-referenced-by-count":0,"title":["The Minimum Spectral Radius of $tP_3$- or $K_5$-Saturated Graphs via the Number of $2$-Walks"],"prefix":"10.37236","volume":"32","author":[{"given":"Jiangdong","family":"Ai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pei","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Suil","family":"O","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junxue","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,2,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p30\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i1p30\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,2,27]],"date-time":"2025-02-27T11:22:41Z","timestamp":1740655361000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i1p30"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,28]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1,17]]}},"URL":"https:\/\/doi.org\/10.37236\/12492","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2025,2,28]]},"article-number":"P1.30"}}