{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,13]],"date-time":"2023-11-13T00:31:22Z","timestamp":1699835482925},"reference-count":10,"publisher":"Wiley","issue":"3","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":8806,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1982,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We propose a generalization of signed graphs, called <jats:italic>group graphs.<\/jats:italic> These are graphs regarded as symmetric digraphs with a group element <jats:italic>s<\/jats:italic>(<jats:italic>u, v<\/jats:italic>) called the <jats:italic>signing<\/jats:italic> associated with each arc (<jats:italic>u, v<\/jats:italic>) such that <jats:italic>s<\/jats:italic>(<jats:italic>u, v<\/jats:italic>) <jats:italic>s<\/jats:italic> (<jats:italic>v, u<\/jats:italic>) = 1. A group graph is <jats:italic>balanced<\/jats:italic> if the product <jats:italic>s<\/jats:italic>(<jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>) <jats:italic>s<\/jats:italic> (<jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>3<\/jats:sub>) \u2026<jats:italic>s<\/jats:italic>(<jats:italic>u<\/jats:italic><jats:sub>m<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>) = 1 for each cycle <jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>2<\/jats:sub>,\u2026, <jats:italic>v<\/jats:italic><jats:sub>m<\/jats:sub>, <jats:italic>v<\/jats:italic><jats:sub>1<\/jats:sub> in the graph. Let <jats:italic>G<\/jats:italic> denote the graph, <jats:italic>F<\/jats:italic> the group (not necessarily commutative), and <jats:italic>s<\/jats:italic> the signing. Then the group graph is denoted by (<jats:italic>G, F, s<\/jats:italic>). Given a group graph (<jats:italic>G, F, s<\/jats:italic>), which need not be balanced, we define the <jats:italic>deletion index<\/jats:italic> \u03b4(<jats:italic>G, F, s<\/jats:italic>) as the minimum cardinality of a <jats:italic>deletion set<\/jats:italic>, which is a subset of lines whose deletion results in a balanced group graph. Similarly we define the <jats:italic>alteration index<\/jats:italic> \u03b3(<jats:italic>G, F, s<\/jats:italic>) as the minimum cardinality of a <jats:italic>alteration set<\/jats:italic>, which is a set of lines {<jats:italic>u, v<\/jats:italic>} in the graph the values <jats:italic>s<\/jats:italic>(<jats:italic>u, v<\/jats:italic>) and <jats:italic>s<\/jats:italic>(<jats:italic>v, u<\/jats:italic>) of which can be changed so that the new group graph is balanced. When <jats:italic>F<\/jats:italic> is the group of order 2, we obtain a <jats:italic>signed graph.<\/jats:italic> Generalizing results known for signed graphs, we prove that minimal deletion sets are minimal change sets, which implies the equality \u03b4(<jats:italic>G, F, s<\/jats:italic>) = \u03b3(<jats:italic>G, F, s<\/jats:italic>) for all <jats:italic>G, F<\/jats:italic>, and <jats:italic>s.<\/jats:italic> We also prove the in\u2010equality \u03b4 (<jats:italic>G, F, s<\/jats:italic>) \u2264 (<jats:italic>n<\/jats:italic> \u2010 1)<jats:italic>q\/n<\/jats:italic>, where <jats:italic>q<\/jats:italic> is the number of lines in the graph <jats:italic>G<\/jats:italic> and <jats:italic>n<\/jats:italic> is the order of the group <jats:italic>F.<\/jats:italic> We conclude by studying \u03b4 for signed complete bigraphs <jats:italic>K<\/jats:italic><jats:sub>n,n<\/jats:sub> when the signing is determined from a Hadamard matrix.<\/jats:p>","DOI":"10.1002\/net.3230120308","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T13:25:15Z","timestamp":1178889915000},"page":"317-321","source":"Crossref","is-referenced-by-count":3,"title":["On balance in group graphs"],"prefix":"10.1002","volume":"12","author":[{"given":"Frank","family":"Harary","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bernt","family":"Lindstr\u00f6m","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hans\u2010olov","family":"Zetterstr\u00f6m","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/bs.3830030102"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1037\/h0046049"},{"key":"e_1_2_1_4_2","first-page":"5","article-title":"Equilibre d'un graphe, quelques r\u00e9sultats alg\u00e9briques","volume":"8","author":"Flament C.","year":"1970","journal-title":"Math. Sci. Humaine"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(76)90059-1"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028989917"},{"key":"e_1_2_1_7_2","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1307\/mmj\/1031710532","article-title":"On local balance and N\u2010balance in signed graphs","volume":"3","author":"Harary F.","year":"1955","journal-title":"Michigan Math. J."},{"key":"e_1_2_1_8_2","doi-asserted-by":"publisher","DOI":"10.1002\/bs.3830040405"},{"key":"e_1_2_1_9_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_10_2","first-page":"123","article-title":"On balance in signed matroids","volume":"6","author":"Harary F.","year":"1981","journal-title":"J. Combinatorics, Information System Sciences"},{"key":"e_1_2_1_11_2","first-page":"37","article-title":"Note sur une caract\u00e9risation des graphes dont le degr\u00e9 de d\u00e9s\u00e9quilibre est maximal","volume":"11","author":"Tomescu I.","year":"1973","journal-title":"Math. Sci. 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