{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T12:34:11Z","timestamp":1772886851179,"version":"3.50.1"},"reference-count":4,"publisher":"Wiley","issue":"4","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":10168,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1978,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A smooth graph is a connected graph without endpoints; <jats:italic>f<\/jats:italic>(<jats:italic>n, q<\/jats:italic>) is the number of connected graphs, <jats:italic>v<\/jats:italic>(<jats:italic>n, q<\/jats:italic>) is the number of smooth graphs, and <jats:italic>u<\/jats:italic>(<jats:italic>n, q<\/jats:italic>) is the number of blocks on <jats:italic>n<\/jats:italic> labeled points and <jats:italic>q<\/jats:italic> edges: <jats:italic>W<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, <jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, and <jats:italic>U<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> are the exponential generating functions of <jats:italic>f<\/jats:italic>(<jats:italic>n, n<\/jats:italic> + <jats:italic>k<\/jats:italic>), <jats:italic>v<\/jats:italic>(<jats:italic>n, n<\/jats:italic> + <jats:italic>k<\/jats:italic>), and <jats:italic>u<\/jats:italic>(<jats:italic>n, n<\/jats:italic> + <jats:italic>k<\/jats:italic>), respectively. For any <jats:italic>k<\/jats:italic> \u2a7e 1, our reduction method shows that <jats:italic>V<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> can be deduced at once from <jats:italic>W<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub>, which was found for successive <jats:italic>k<\/jats:italic> by the computer method described in our previous paper. Again the reduction method shows that <jats:italic>U<\/jats:italic><jats:sub>k<\/jats:sub> must be a sum of powers (mostly negative) of 1 \u2010 <jats:italic>X<\/jats:italic> and, given this information, we develop a recurrence method well suited to calculate <jats:italic>U<\/jats:italic><jats:sub><jats:italic>k<\/jats:italic><\/jats:sub> for successive <jats:italic>k<\/jats:italic>. Exact formulas for <jats:italic>v<\/jats:italic>(<jats:italic>n, n<\/jats:italic> + <jats:italic>k<\/jats:italic>) and <jats:italic>u<\/jats:italic>(<jats:italic>n, n<\/jats:italic> + <jats:italic>k<\/jats:italic>) for general <jats:italic>n<\/jats:italic> follow at once.<\/jats:p>","DOI":"10.1002\/jgt.3190020403","type":"journal-article","created":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T10:17:27Z","timestamp":1180174647000},"page":"299-305","source":"Crossref","is-referenced-by-count":38,"title":["The number of connected sparsely edged graphs. II. Smooth graphs and blocks"],"prefix":"10.1002","volume":"2","author":[{"given":"E. M.","family":"Wright","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.1698742"},{"key":"e_1_2_1_3_2","unstructured":"C. C.Rousseau A note on the enumeration of connected graphs. To appear."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1088\/0370-1328\/72\/6\/424"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190010407"}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190020403","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190020403","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T10:59:53Z","timestamp":1699786793000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190020403"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1978,12]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1978,12]]}},"alternative-id":["10.1002\/jgt.3190020403"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190020403","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1978,12]]}}}