{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,23]],"date-time":"2025-10-23T09:35:27Z","timestamp":1761212127131,"version":"build-2065373602"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Math. Algorithm. Appl."],"published-print":{"date-parts":[[2025,11]]},"abstract":"<jats:p>A graph [Formula: see text] is [Formula: see text]-integral if its signless Laplacian has only integral eigenvalues. Suppose [Formula: see text] be the graph after adding a self-loop to every vertex of a subset [Formula: see text] of [Formula: see text]. In this paper, we prove that for a nonempty proper subset [Formula: see text] of [Formula: see text], [Formula: see text] is [Formula: see text]-integral if and only if [Formula: see text] and self-loops are added in exactly two vertices of each partite set, or [Formula: see text] and self-loops are added to all the vertices of one partite set. Besides, we prove that if [Formula: see text] is a nonempty proper subset of [Formula: see text], then [Formula: see text] is [Formula: see text]-integral if and only if [Formula: see text] and self-loops are added to all the pendant vertices. Using these results, we also answer when the matrix [Formula: see text] have only integral spectrum, where [Formula: see text] is any [Formula: see text]-diagonal matrix and [Formula: see text] is permutation similar to either [Formula: see text] or [Formula: see text].<\/jats:p>","DOI":"10.1142\/s1793830924501258","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T21:14:12Z","timestamp":1731532452000},"source":"Crossref","is-referenced-by-count":0,"title":["A note on the Q-integrability of complete bipartite graphs with self-loops"],"prefix":"10.1142","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1276-1515","authenticated-orcid":false,"given":"Jesmina","family":"Pervin","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh 221005, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8208-5372","authenticated-orcid":false,"given":"Lavanya","family":"Selvaganesh","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Indian Institute of Technology (BHU) Varanasi, Uttar Pradesh 221005, India"}]}],"member":"219","published-online":{"date-parts":[[2024,12,23]]},"reference":[{"key":"S1793830924501258BIB001","doi-asserted-by":"publisher","DOI":"10.46793\/match.90-1.247V"},{"key":"S1793830924501258BIB002","first-page":"42","volume":"13","author":"Bali\u0144ska K.","year":"2002","journal-title":"Publ. 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