{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,11]],"date-time":"2025-06-11T09:31:35Z","timestamp":1749634295479},"reference-count":13,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T00:00:00Z","timestamp":1616025600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T00:00:00Z","timestamp":1616025600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2022,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A conference matrix of order <jats:italic>n<\/jats:italic> is an <jats:inline-formula><jats:alternatives><jats:tex-math>$$n\\times n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> matrix <jats:italic>C<\/jats:italic> with diagonal entries 0 and off-diagonal entries <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pm 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>\u00b1<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfying <jats:inline-formula><jats:alternatives><jats:tex-math>$$CC^\\top =(n-1)I$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>C<\/mml:mi>\n                      <mml:mi>\u22a4<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mi>I<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. If <jats:italic>C<\/jats:italic> is symmetric, then <jats:italic>C<\/jats:italic> has a symmetric spectrum <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03a3<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (that is, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\Sigma =-\\Sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03a3<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mi>\u03a3<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) and eigenvalues <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\pm \\sqrt{n-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>\u00b1<\/mml:mo>\n                    <mml:msqrt>\n                      <mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msqrt>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We show that many principal submatrices of <jats:italic>C<\/jats:italic> also have symmetric spectrum, which leads to examples of Seidel matrices of graphs (or, equivalently, adjacency matrices of complete signed graphs) with a symmetric spectrum. In addition, we show that some Seidel matrices with symmetric spectrum can be characterized by this construction.<\/jats:p>","DOI":"10.1007\/s10623-021-00858-8","type":"journal-article","created":{"date-parts":[[2021,3,18]],"date-time":"2021-03-18T21:44:51Z","timestamp":1616103891000},"page":"1983-1990","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Spectral symmetry in conference matrices"],"prefix":"10.1007","volume":"90","author":[{"given":"Willem H.","family":"Haemers","sequence":"first","affiliation":[]},{"given":"Leila","family":"Parsaei Majd","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,3,18]]},"reference":[{"key":"858_CR1","doi-asserted-by":"crossref","unstructured":"Akbari S., Maimani H.R., Parsaei Majd L.: On the spectrum of some signed complete and complete bipartite graphs. 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Springer, New York (2012)."},{"key":"858_CR5","doi-asserted-by":"crossref","unstructured":"Bussemaker F.C., Mathon R., Seidel J.J.: Tables of two-graphs, Combinatorics and graph theory (S.B. Rao ed.). Springer, Berlin (Lecture Notes in Math. 885), pp. 70\u2013112 (1981).","DOI":"10.1007\/BFb0092256"},{"key":"858_CR6","doi-asserted-by":"crossref","unstructured":"Ghorbani E., Haemers W.H., Maimani H.R., Parsaei Majd L.: On sign-symmetric signed graphs. Ars Math. Contemporanea 19, 83\u201393 (2020). also: arXiv:2003.09981.","DOI":"10.26493\/1855-3974.2161.f55"},{"key":"858_CR7","doi-asserted-by":"publisher","first-page":"208","DOI":"10.1016\/j.jcta.2015.09.008","volume":"138","author":"GRW Greaves","year":"2016","unstructured":"Greaves G.R.W., Koolen J.H., Munemasa A., Sz\u00f6ll\u0151si F.: Equiangular lines in Euclidean spaces. J. Comb. Theory Ser. A 138, 208\u2013235 (2016).","journal-title":"J. Comb. Theory Ser. 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