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Math."],"published-print":{"date-parts":[[2022,4]]},"abstract":"Abstract<\/jats:title>Let $$\\dot{G}=(G,\\sigma )$$<\/jats:tex-math>\n \n \n G<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n =<\/mml:mo>\n \n (<\/mml:mo>\n G<\/mml:mi>\n ,<\/mml:mo>\n \u03c3<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a signed graph, and let $$\\rho (\\dot{G})$$<\/jats:tex-math>\n \n \u03c1<\/mml:mi>\n (<\/mml:mo>\n \n G<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (resp.\u00a0$$\\lambda _1(\\dot{G})$$<\/jats:tex-math>\n \n \n \u03bb<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n G<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) denote the spectral radius (resp.\u00a0the index) of the adjacency matrix $$A_{\\dot{G}}$$<\/jats:tex-math>\n \n A<\/mml:mi>\n \n G<\/mml:mi>\n \u02d9<\/mml:mo>\n <\/mml:mover>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper we detect the signed graphs achieving the minimum spectral radius $$m(\\mathcal S \\mathcal R_n)$$<\/jats:tex-math>\n \n m<\/mml:mi>\n (<\/mml:mo>\n S<\/mml:mi>\n \n R<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the maximum spectral radius $$M(\\mathcal S \\mathcal R_n)$$<\/jats:tex-math>\n \n M<\/mml:mi>\n (<\/mml:mo>\n S<\/mml:mi>\n \n R<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the minimum index $$m(\\mathcal I_n)$$<\/jats:tex-math>\n \n m<\/mml:mi>\n (<\/mml:mo>\n \n I<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the maximum index $$M(\\mathcal I_n)$$<\/jats:tex-math>\n \n M<\/mml:mi>\n (<\/mml:mo>\n \n I<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in the set $$\\mathcal U_n$$<\/jats:tex-math>\n \n U<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of all unbalanced connected signed graphs with $$n\\geqslant 3$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2a7e<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> vertices. From the explicit computation of the four extremal values it turns out that the difference $$m(\\mathcal S \\mathcal R_n)-m(\\mathcal I_n)$$<\/jats:tex-math>\n \n m<\/mml:mi>\n \n (<\/mml:mo>\n S<\/mml:mi>\n \n R<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n -<\/mml:mo>\n m<\/mml:mi>\n \n (<\/mml:mo>\n \n I<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for $$n \\geqslant 8$$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2a7e<\/mml:mo>\n 8<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> strictly increases with n<\/jats:italic> and tends to 1, whereas $$M(\\mathcal S \\mathcal R_n)- M(\\mathcal I_n)$$<\/jats:tex-math>\n \n M<\/mml:mi>\n \n (<\/mml:mo>\n S<\/mml:mi>\n \n R<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n -<\/mml:mo>\n M<\/mml:mi>\n \n (<\/mml:mo>\n \n I<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> strictly decreases and tends to 0.<\/jats:p>","DOI":"10.1007\/s40314-022-01814-5","type":"journal-article","created":{"date-parts":[[2022,3,25]],"date-time":"2022-03-25T04:39:15Z","timestamp":1648183155000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Unbalanced signed graphs with extremal spectral radius or index"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2742-1919","authenticated-orcid":false,"given":"Maurizio","family":"Brunetti","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4949-4203","authenticated-orcid":false,"given":"Zoran","family":"Stani\u0107","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2022,3,24]]},"reference":[{"key":"1814_CR1","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1016\/j.laa.2018.05.012","volume":"553","author":"S Akbari","year":"2018","unstructured":"Akbari S, Belardo F, Dodongeh E, Nematollahi MA (2018) Spectral characterizations of signed cycles. 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