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Here it is shown that such a theory for the category of graphs that do not allow loops degenerates to the trivial case for all the Hoehnke radicals, but there are non-trivial connectednesses (KA-radical classes) and disconnectednesses (KA-semisimple classes). Moreover, the connectednesses and disconnectednesses always come as complementary pairs.\n<\/jats:p>","DOI":"10.1007\/s10998-022-00496-0","type":"journal-article","created":{"date-parts":[[2022,8,13]],"date-time":"2022-08-13T11:02:42Z","timestamp":1660388562000},"page":"27-39","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A radical theory for graphs that do not admit loops"],"prefix":"10.1007","volume":"87","author":[{"given":"Stefan","family":"Veldsman","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,8,13]]},"reference":[{"key":"496_CR1","doi-asserted-by":"publisher","first-page":"764","DOI":"10.1090\/S0002-9904-1944-08235-9","volume":"50","author":"G Birkhoff","year":"1944","unstructured":"G. Birkhoff, Subdirect unions in universal algebra. Bull. Am. Math. Soc. 50, 764\u2013768 (1944)","journal-title":"Bull. Am. Math. 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