{"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":1776965713665,"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 polynomial reconstruction problem, proposed by Cvetkovi\u0107 in 1973, asks whether the characteristic polynomial $\\phi(G;x)$ of a graph $G$ with at least 3 vertices can be reconstructed from the polynomial deck $\\mathcal{P}(G)=\\{\\phi(G-v_i;x)\\}_{v_i\\in V(G)}$. In 2000, Hagos proposed a question of whether $\\phi(G;x)$ is reconstructible from the two decks $\\mathcal{P}(G)$ and $\\mathcal{P}(\\bar{G})$. Recently, strengthening a theorem of Ji et al. (2024), Spier (2025) proved that the characteristic polynomials of two graphs are congruent modulo 4 if and only if the characteristic polynomials of their complements are congruent modulo 4. Motivated by the above, we prove that for any two graphs $G$ and $H$, if $\\phi(G;x)-\\phi(H;x)\\equiv c\\pmod{4}$ and $\\phi(\\bar{G};x)-\\phi(\\bar{H};x)\\equiv d\\pmod{4}$ for two constants $c$ and $d$, respectively, then $c\\equiv d\\pmod{4}$. In particular, if $n$ is even, then $c\\equiv d\\equiv 0\\pmod{4}$. This strengthens Spier's results, and provides some non-trivial information about the constant coefficients of $\\phi(G;x)$ and $\\phi(H;x)$ for any potential counterexample pair $(G,H)$ to the polynomial reconstruction problem. We also obtain a similar result for any potential counterexample pair $(G,H)$ to the problem proposed by Hagos in 2000.<\/jats:p>","DOI":"10.37236\/14650","type":"journal-article","created":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T16:38:34Z","timestamp":1776962314000},"source":"Crossref","is-referenced-by-count":0,"title":["Nearly Cospectral Graphs and the Polynomial Reconstruction Problem"],"prefix":"10.37236","volume":"33","author":[{"given":"Weifang","family":"Lv","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hao","family":"Zhang","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\/v33i2p14\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i2p14\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T16:38:34Z","timestamp":1776962314000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i2p14"}},"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\/14650","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,24]]},"article-number":"P2.14"}}