{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T18:47:28Z","timestamp":1776451648886,"version":"3.51.2"},"reference-count":8,"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":10533,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1977,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>An (<jats:italic>n, q<\/jats:italic>) graph has <jats:italic>n<\/jats:italic> labeled points, <jats:italic>q<\/jats:italic> edges, and no loops or multiple edges. The number of connected (<jats:italic>n, q<\/jats:italic>) graphs is <jats:italic>f(n, q)<\/jats:italic>. Cayley proved that <jats:italic>f(n, n<\/jats:italic><jats:sup>\u20101<\/jats:sup>) = <jats:italic>n<\/jats:italic><jats:sup>n\u22122<\/jats:sup> and Renyi found a formula for <jats:italic>f(n, n)<\/jats:italic>. Here I develop two methods to calculate the exponential generating function of <jats:italic>f(n, n + k)<\/jats:italic> for particular <jats:italic>k<\/jats:italic> and so to find a formula for <jats:italic>f(n, n + k)<\/jats:italic> for general <jats:italic>n.<\/jats:italic> The first method is a recurrent one with respect to <jats:italic>k<\/jats:italic> and is well adapted for machine computation, but does not itself provide a proof that it can be continued indefinitely. The second (reduction) method is much less efficient and is indeed impracticable for <jats:italic>k<\/jats:italic> greater than 2 or 3, but it supplies the missing proof that the generating function is of a particular form and so that the first method can be continued for all <jats:italic>k<\/jats:italic>, subject only to the capacity of the machine.<\/jats:p>","DOI":"10.1002\/jgt.3190010407","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:10:00Z","timestamp":1180419000000},"page":"317-330","source":"Crossref","is-referenced-by-count":86,"title":["The number of connected sparsely edged graphs"],"prefix":"10.1002","volume":"1","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":"10.1007\/BF02948945"},{"key":"e_1_2_1_3_2","first-page":"376","article-title":"A theorem on trees","volume":"23","author":"Cayley A.","year":"1889","journal-title":"Quart. J. Math."},{"key":"e_1_2_1_4_2","first-page":"64","article-title":"Eine Formel der Substitutionstheorie","volume":"17","author":"Dziobek O.","year":"1917","journal-title":"Sitzungsber. Berliner Math. Gesellsch."},{"key":"e_1_2_1_5_2","doi-asserted-by":"crossref","first-page":"215","DOI":"10.21236\/AD0705364","volume-title":"Graph Theory","author":"Harary F.","year":"1969"},{"key":"e_1_2_1_6_2","first-page":"141","volume-title":"Funktionentheorie","author":"Hurwitz A.","year":"1929"},{"key":"e_1_2_1_7_2","unstructured":"J. W.Moon Various proofs of Cayley's formula for counting trees.A seminar on graph theory edited by F. Harary. New York (1967)70\u201378."},{"issue":"159","key":"e_1_2_1_8_2","first-page":"385","article-title":"On connected graphs I","volume":"4","author":"Renyi A.","journal-title":"Publ. Math. Inst. Hungarian Acad. Sci."},{"key":"e_1_2_1_9_2","first-page":"193","article-title":"Solution of the equation zez\n                   = a","volume":"65","author":"Wright E. M.","year":"1959","journal-title":"Proc. Roy. Soc. Edinburgh."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190010407","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190010407","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,12]],"date-time":"2023-11-12T09:52:27Z","timestamp":1699782747000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190010407"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1977,12]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1977,12]]}},"alternative-id":["10.1002\/jgt.3190010407"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190010407","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1977,12]]}}}