{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,18]],"date-time":"2026-07-18T11:20:09Z","timestamp":1784373609254,"version":"3.55.0"},"reference-count":90,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,11]],"date-time":"2006-10-11T00:00:00Z","timestamp":1160524800000},"content-version":"vor","delay-in-days":7894,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1985,3]]},"DOI":"10.1002\/net.3230150108","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T19:18:17Z","timestamp":1178911097000},"page":"73-107","source":"Crossref","is-referenced-by-count":22,"title":["A compilation of relations between graph invariants"],"prefix":"10.1002","volume":"15","author":[{"given":"Robert C.","family":"Brigham","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ronald D.","family":"Dutton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02020809"},{"key":"e_1_2_1_3_2","unstructured":"M.Ajtai V.Chv\u00e1tal M. M.NewbornandE.Szemer\u00e9di Crossing\u2010free subgraphs. Technical Report SOCS\u201079\u201021 School of Computer Science McGill University Montreal February 1981 (Annals Discrete Math in press.)"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(74)90133-2"},{"issue":"143","key":"e_1_2_1_5_2","first-page":"212","article-title":"The thickness of an arbitrary complete graph","volume":"101","author":"Alekseev V. B.","year":"1976","journal-title":"Mat. Sb. (N. S.)"},{"key":"e_1_2_1_5_3","doi-asserted-by":"publisher","DOI":"10.1070\/SM1976v030n02ABEH002267"},{"key":"e_1_2_1_6_2","doi-asserted-by":"publisher","DOI":"10.1137\/0122052"},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0059416"},{"key":"e_1_2_1_8_2","first-page":"29","volume-title":"Theory of Graphs","author":"Bollob\u00e1s B.","year":"1968"},{"key":"e_1_2_1_9_2","first-page":"25","volume-title":"Combinatorial Mathematics and its Applications","author":"Bollob\u00e1s B.","year":"1969"},{"key":"e_1_2_1_10_2","volume-title":"External Graph Theory","author":"Bollob\u00e1s B.","year":"1978"},{"key":"e_1_2_1_11_2","unstructured":"J. C.Bermond andB.Bollab\u00e1s The diameter of graphs\u2010a survey Proc. 12th Southeastern Conf. on Combinatorics Graph Theory and Computing (Louisiana State University Baton Rouge 1981); Congressus Numerantium Vol. 32. Utilitas Math Winnipeg (1981) pp.3\u201327."},{"key":"e_1_2_1_12_2","doi-asserted-by":"crossref","unstructured":"J. M.Benedict andP. Z.Chinn On graphs having prescribed clique number chromatic number and maximum degree. Theory and Application of Graphs (Proc. Internat. Conf. Western Michigan University Kalamazoo Michigan 1976) Lecture Notes in Math 642. Springer Berlin (1978) pp.132\u2013140.","DOI":"10.1007\/BFb0070370"},{"key":"e_1_2_1_13_2","volume-title":"Graphs and Digraphs","author":"Behzad M.","year":"1979"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(81)90016-9"},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(83)90054-7"},{"key":"e_1_2_1_16_2","unstructured":"R. C.Brigham andR. D.Dutton Upper bounds on the edge clique cover number of a graph. Discrete Math. (in press)."},{"key":"e_1_2_1_17_2","unstructured":"B.Bollob\u00e1s andS. E.Eldridge Maximal matchings in graphs with given maximal and minimal degrees. Proc. 5th British Combinatorial Conf. (University Aberdeen Aberdeen 1975). Congressus Numerantium No. XV. Utilitas Math. Winnipeg Manitoba (1976) pp.165\u2013168."},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1017\/S2040618500035097"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(77)90037-5"},{"key":"e_1_2_1_20_2","first-page":"239","article-title":"Extremal graphs of diameter two with prescribed minimum degree","volume":"7","author":"Bondy J. A.","year":"1972","journal-title":"Studio Sci. Math. Hungar."},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(78)90076-X"},{"key":"e_1_2_1_22_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(71)90019-7"},{"key":"e_1_2_1_23_2","unstructured":"F.Buckley Self\u2010centered graphs with a given radius. Proc. 10th Southeastern Conference on Combinatorics Graph Theory and Computing (Florida Atlantic Univ. Boca Raton Florida 1979). Congressus Numerantium XXIII. Utilitas Winnipeg (1979) pp.211\u2013215."},{"issue":"28","key":"e_1_2_1_24_2","first-page":"25","article-title":"Chromatic number and the spectrum of a graph","volume":"14","author":"Cvetkovi\u0107 D. M.","year":"1972","journal-title":"Publ. Inst. Math (Beograd) (N. S.)"},{"key":"e_1_2_1_25_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(78)90167-X"},{"key":"e_1_2_1_26_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190060302"},{"key":"e_1_2_1_27_2","volume-title":"Spectra of Graphs","author":"Cvetkovi\u0107 D. M.","year":"1980"},{"key":"e_1_2_1_28_2","doi-asserted-by":"publisher","DOI":"10.1111\/j.1749-6632.1979.tb32840.x"},{"key":"e_1_2_1_29_2","doi-asserted-by":"publisher","DOI":"10.1137\/0114065"},{"key":"e_1_2_1_30_2","unstructured":"V.Chv\u00e1tal New directions in Hamiltonian graph theory. New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf. on Graph Theory). University of Michigan Ann Arbor (1971) pp.65\u201395."},{"key":"e_1_2_1_31_2","doi-asserted-by":"crossref","first-page":"366","DOI":"10.21136\/CMJ.1971.101034","article-title":"Some relations among invariants of graphs","volume":"21","author":"Chv\u00e1tal V.","year":"1971","journal-title":"Czecho Math J."},{"key":"e_1_2_1_32_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(76)90004-6"},{"key":"e_1_2_1_33_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(70)80057-6"},{"key":"e_1_2_1_34_2","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s1-44.1.612"},{"key":"e_1_2_1_35_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760181"},{"key":"e_1_2_1_36_2","volume-title":"Examples and Counterexamples in Graph Theory","author":"Capobianco M.","year":"1978"},{"key":"e_1_2_1_37_2","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/s2-8.2.322"},{"key":"e_1_2_1_38_2","doi-asserted-by":"publisher","DOI":"10.2307\/2040674"},{"key":"e_1_2_1_39_2","doi-asserted-by":"publisher","DOI":"10.1112\/S0025579300004770"},{"key":"e_1_2_1_40_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02018401"},{"key":"e_1_2_1_41_2","doi-asserted-by":"crossref","unstructured":"R. J.Cook Analogues of Heawood's theorem. Combinatorics (Proc. British Combinatorial Conf.). Univ. Coll. Wales Aberystwyth (1973) pp.27\u201333.","DOI":"10.1017\/CBO9780511662072.006"},{"key":"e_1_2_1_42_2","volume-title":"Theory and Application of Graphs in America's Bicentennial Year","author":"Cockayne E. J.","year":"1976"},{"key":"e_1_2_1_43_2","first-page":"289","article-title":"Line\u2010clique cover number of a graph","volume":"41","author":"Choudum S. A.","year":"1975","journal-title":"Prod. Ind. Nat. Sci. Acad."},{"key":"e_1_2_1_44_2","first-page":"433","article-title":"Inequalities obtained on the basis of the spectrum of the graph","volume":"8","author":"Cvetkovi\u0107 D. M.","year":"1973","journal-title":"Stud. Sci. Math. Hungar."},{"key":"e_1_2_1_45_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(72)90003-9"},{"key":"e_1_2_1_46_2","unstructured":"A. K.Dewdney The bandwidth of a graph\u2010some recent results. Proc. 7th Southeastern Conference on Combinatorics Graph Theory and Computing (Louisiana State University Baton Rouge LA 1976). Congressus Numerantium XVII. Utilitas (1976) pp.273\u2013278."},{"key":"e_1_2_1_47_2","first-page":"59","article-title":"Valency\u2010variety and chromatic number of abstract graphs","volume":"13","author":"Dirac G. A.","year":"1964","journal-title":"Wissen\u2010schaftliche Zeitschrift der Martin\u2010Luther Universit\u00e4t Halle\u2010Wittenberg"},{"key":"e_1_2_1_48_2","doi-asserted-by":"crossref","first-page":"290","DOI":"10.21136\/CPM.1969.108598","article-title":"On the number of complete subgraphs and circuits contained in graphs","volume":"94","author":"Erd\u00f6s P.","year":"1969","journal-title":"\u010casopis Pro P\u011bstov\u0301ani Matematiky"},{"key":"e_1_2_1_49_2","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100052233"},{"key":"e_1_2_1_50_2","doi-asserted-by":"publisher","DOI":"10.1016\/0166-218X(83)90015-X"},{"key":"e_1_2_1_51_2","first-page":"181","article-title":"On the minimal number of vertices representing the edges of a graph","volume":"6","author":"Erd\u00f6s P.","year":"1961","journal-title":"Akad. Mat. Intezenck Kozlemanyai"},{"key":"e_1_2_1_52_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1966-014-3"},{"key":"e_1_2_1_53_2","doi-asserted-by":"crossref","unstructured":"S.Fiorini andR. J.Wilson On the chromatic index of a graph II. Combinatorics (Proc. British Combinatorial Conf. Univ. Coll. Wales Aberystwyth 1973) London Math Soc. Lecture Note Ser. No. 13. Cambridge Univ. Press London (1974) pp.37\u201351.","DOI":"10.1017\/CBO9780511662072.008"},{"key":"e_1_2_1_54_2","first-page":"99","article-title":"Maximal graphs with given connectivity and edge\u2010connectivity. Mat","volume":"25","author":"Gliviak F.","year":"1975","journal-title":"Casopis (Slovenska Academia Vied)"},{"key":"e_1_2_1_55_2","unstructured":"J. R.Griggs Lower bounds on the independence number in terms of the degress. Presented at 793rd Meeting of the American Mathematical Society Bryn Mawr College Bryn Mawr Pennsylvania March 16\u201017 (1982)."},{"key":"e_1_2_1_56_2","doi-asserted-by":"publisher","DOI":"10.21236\/AD0705364"},{"key":"e_1_2_1_57_2","unstructured":"P.Hansen Degres et nombre de stabilit\u00e9 d'un graphe. Colloque Sur La Theorie Des Graphes (Paris 1974). Cahiers Centre Etudes Recherche Oper. V. 17 (1975)."},{"key":"e_1_2_1_58_2","first-page":"1195","article-title":"Upper bounds for the stability number of a graph","volume":"24","author":"Hansen P.","year":"1979","journal-title":"Rev. Roum. Math. Pures Appl."},{"key":"e_1_2_1_59_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190020412"},{"key":"e_1_2_1_60_2","doi-asserted-by":"crossref","unstructured":"H. V.Kronk The chromatic number of triangle\u2010free graphs Graph Theory and Applications (Proc. Conf. Western Michigan University Kalamazoo Mich. 1972). Lecture Notes in Math. Vol. 303. Springer Berlin (1972) pp.179\u2013181.","DOI":"10.1007\/BFb0067369"},{"key":"e_1_2_1_61_2","doi-asserted-by":"crossref","unstructured":"H. V.Kronk Variations on a theorem of P\u00f3sa. The Many Facets of Graph Theory (Proc. Conf. Western Michigan University Kalamazoo Mich. 1968). Springer Berlin (1969) pp.193\u2013197.","DOI":"10.1007\/BFb0060117"},{"key":"e_1_2_1_62_2","first-page":"17","article-title":"On the edge\u2010chromatic number of a graph, Reports of the faculty of science and engineering","volume":"3","author":"Kitamura S.","year":"1975","journal-title":"Saga Univ. Math."},{"key":"e_1_2_1_63_2","doi-asserted-by":"publisher","DOI":"10.2307\/2042566"},{"key":"e_1_2_1_64_2","first-page":"9","article-title":"On nonplanar graphs with the minimum number of vertices and a given girth","volume":"11","author":"Koman M.","year":"1970","journal-title":"Kommentationes Math. Univ. Carolinae (Prague)"},{"key":"e_1_2_1_65_2","first-page":"53","article-title":"On the stability number of a graph","volume":"10","author":"Koh K. M.","year":"1977","journal-title":"Nanta Math."},{"key":"e_1_2_1_66_2","first-page":"957","article-title":"Upper bounds on the chromatic number of a graph in terms of its degree, density and girth","volume":"18","author":"Kost\u01d2cka A. V.","year":"1977","journal-title":"Soviet Math. Dokl."},{"key":"e_1_2_1_67_2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1.1.28"},{"key":"e_1_2_1_68_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(76)90031-5"},{"key":"e_1_2_1_69_2","doi-asserted-by":"crossref","unstructured":"D. R.Lick A class of point partition numbers. Recent Trends in Graph Theory (Proc. Conf. New York 1970). Lecture Notes in Mathematics Vol. 186. Springer Berlin (1971) pp.185\u2013190.","DOI":"10.1007\/BFb0059436"},{"key":"e_1_2_1_70_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(75)90089-1"},{"key":"e_1_2_1_71_2","unstructured":"L.Lov\u00e1sz On covering of graphs. Theory of Graphs (Proc. Colloq. Tihany 1966). Academic Press New York (1968) pp.231\u2013236."},{"key":"e_1_2_1_72_2","volume-title":"Combinatorial Problems and Exercises","author":"Lov\u00e1sz L.","year":"1979"},{"key":"e_1_2_1_73_2","series-title":"Lecture Notes in Math. 885","first-page":"308","volume-title":"On domination related concepts in graph theory","author":"Laskar R. C.","year":"1980"},{"key":"e_1_2_1_74_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(77)90162-5"},{"key":"e_1_2_1_75_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(77)90033-8"},{"key":"e_1_2_1_76_2","doi-asserted-by":"publisher","DOI":"10.1002\/net.3230010306"},{"key":"e_1_2_1_77_2","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1028999370"},{"key":"e_1_2_1_78_2","doi-asserted-by":"publisher","DOI":"10.1016\/0095-8956(77)90033-8"},{"key":"e_1_2_1_79_2","doi-asserted-by":"publisher","DOI":"10.2307\/2306658"},{"key":"e_1_2_1_80_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(67)80036-X"},{"key":"e_1_2_1_81_2","first-page":"704","article-title":"Sul numero minimo degli spigoli di un singramma di raggio e diametro equali a due","volume":"106","author":"Palumbiny D.","year":"1972","journal-title":"Accademia di Scienze e Lettre, Institute Lombardo (Rendiconti Scienze Matematiche)"},{"key":"e_1_2_1_82_2","first-page":"71","article-title":"Critical graphs of given diameter","author":"Plesnik J.","year":"1975","journal-title":"Acta Facultatis Rerum Naturalium Universitatis Comenianae Mathematica"},{"key":"e_1_2_1_83_2","doi-asserted-by":"crossref","unstructured":"R. D.Ringeisen Upper and lower imbeddable graphs. Graph Theory and Applications (Proc. of Conf. at Western Michigan Univ. May 10\u201013 1972). Lecture Notes in Math. Vol. 303. Springer Berlin (1972) pp.261\u2013268.","DOI":"10.1007\/BFb0067378"},{"key":"e_1_2_1_84_2","unstructured":"N.Sauer The largest number of edges of a graph such that not more than g intersect in a point or more than n are independent. Combinatorial Mathematics and its Applications (Proc. Conf. Oxford 1969). Academic Press London (1971) pp.253\u2013257."},{"key":"e_1_2_1_85_2","doi-asserted-by":"publisher","DOI":"10.2307\/2315106"},{"key":"e_1_2_1_86_2","doi-asserted-by":"publisher","DOI":"10.1007\/BF02576041"},{"key":"e_1_2_1_87_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0012-365X(76)80010-6"},{"key":"e_1_2_1_88_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-7557-9_17"},{"key":"e_1_2_1_89_2","first-page":"159","article-title":"Upper bound to the number of edges of a graph with specified nondensity and all\u2010contiguity numbers","volume":"1","author":"Vinnichenko N. G.","year":"1972","journal-title":"Kibernetika (Kiev)"},{"key":"e_1_2_1_90_2","doi-asserted-by":"publisher","DOI":"10.2307\/2316031"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230150108","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230150108","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T15:44:44Z","timestamp":1697816684000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230150108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1985,3]]},"references-count":90,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1985,3]]}},"alternative-id":["10.1002\/net.3230150108"],"URL":"https:\/\/doi.org\/10.1002\/net.3230150108","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1985,3]]}}}