{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T11:42:31Z","timestamp":1697888551764},"reference-count":20,"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":7529,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Networks"],"published-print":{"date-parts":[[1986,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Part of this work is concerned with determining those graphs that produce extreme rays of the cone of shortest distance matrices of undirected linear <jats:italic>n<\/jats:italic>\u2010vertex graphs without loops. For <jats:italic>n<\/jats:italic> \u2264 4, for example, all extreme rays come from a certain class of \u201celementary\u201d graphs, but for <jats:italic>n<\/jats:italic> \u2265 5 there are nonelementary extreme rays, and these are studied further. Another part of this work is concerned with the representation of a distance matrix as a set of distances among points in a normed vector space. The set of graphs (distance matrices) representable by a given norm and the set of norms (and dimensions of spaces) by which a given graph may be represented are studied. It is shown that l<jats:sub>x<\/jats:sub> is unique among l<jats:sub>p<\/jats:sub> norms in that any graph is l<jats:sub>x<\/jats:sub> representable in R<jats:sup><jats:italic>n<\/jats:italic>\u22121<\/jats:sup>. The l<jats:sub>1<\/jats:sub> representable distance matrices are found to be just those in the cone generated by elementaries for all <jats:italic>n<\/jats:italic>. Other results on distance matrices are also included.<\/jats:p>","DOI":"10.1002\/net.3230160109","type":"journal-article","created":{"date-parts":[[2007,5,11]],"date-time":"2007-05-11T20:21:48Z","timestamp":1178914908000},"page":"87-106","source":"Crossref","is-referenced-by-count":6,"title":["The structure of distances in networks"],"prefix":"10.1002","volume":"16","author":[{"given":"Eric","family":"Howe","sequence":"first","affiliation":[]},{"given":"Charles R.","family":"Johnson","sequence":"additional","affiliation":[]},{"given":"James","family":"Lawrence","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,11]]},"reference":[{"key":"e_1_2_1_2_2","unstructured":"P.AssouadandM.Deza Isometric embedding in L1 in hypercubes and related problems. Manuscript 1979."},{"key":"e_1_2_1_3_2","unstructured":"D.Avis Hypermetric spaces and the hamming cone. Manuscript 1979."},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1980-010-0"},{"key":"e_1_2_1_5_2","volume-title":"Theory and Applications of Distance Geometry","author":"Blumenthal L. M.","year":"1953"},{"key":"e_1_2_1_6_2","unstructured":"P. M.DearingandG. J.Langford The multifacility total cost location problem on a tree network. Technical Report # 209 Bell Laboratories Holmdel NJ 1975."},{"key":"e_1_2_1_7_2","volume-title":"Communication, Transmission, and Transportation Networks","author":"Frank H.","year":"1971"},{"key":"e_1_2_1_8_2","first-page":"153","article-title":"Realizing the distance matrix of a graph","volume":"70","author":"Goldman A. J.","year":"1966","journal-title":"J. Res. NBS"},{"key":"e_1_2_1_9_2","unstructured":"R. L.Graham A.Yao andF.Yao Information bounds are weak in the shortest distance problem. J. ACM in press."},{"key":"e_1_2_1_10_2","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/qam\/414405","article-title":"The distance matrix of a graph and its tree realizability","volume":"30","author":"Hakimi S. L.","year":"1972","journal-title":"Q. Appl. Math."},{"key":"e_1_2_1_11_2","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/184873"},{"issue":"2","key":"e_1_2_1_12_2","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1112\/jlms\/2.Part_4.727","article-title":"Graphical representations of complete metric spaces","volume":"2","author":"Holmes R. A.","year":"1970","journal-title":"J. London Math. Soc."},{"key":"e_1_2_1_13_2","unstructured":"W.Imrich On metric properties of tree\u2010like spaces. I. Beitrage zur Graphentheorie und deren Anwendungen vorgetragen auf den internationalen Kolloquium in Oberhof (DDR) 1977 129\u2013156."},{"key":"e_1_2_1_14_2","unstructured":"J. D.Murchland A fixed matrix method for all shortest distances in a directed graph and for the inverse problem. Dissertation Karlsruhe 1970."},{"key":"e_1_2_1_15_2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1938-1501980-0"},{"key":"e_1_2_1_16_2","unstructured":"P. D.Seymour Sums of circuits. Graph Theory Rel. Top. (1979)341\u2013355."},{"key":"e_1_2_1_17_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(69)80092-X"},{"key":"e_1_2_1_18_2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-46216-0"},{"key":"e_1_2_1_19_2","doi-asserted-by":"publisher","DOI":"10.1016\/S0021-9800(67)80058-9"},{"key":"e_1_2_1_20_2","first-page":"136","article-title":"Imbedding a finite metric set in an N\u2010dimensional Minkowski space","volume":"70","author":"Wolfe D.","year":"1967","journal-title":"Proceedings of koninklijke Nederlandse Akademie van Wetenschappen"},{"key":"e_1_2_1_21_2","doi-asserted-by":"publisher","DOI":"10.2307\/2039412"}],"container-title":["Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fnet.3230160109","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/net.3230160109","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,21]],"date-time":"2023-10-21T01:51:01Z","timestamp":1697853061000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/net.3230160109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1986,3]]},"references-count":20,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1986,3]]}},"alternative-id":["10.1002\/net.3230160109"],"URL":"https:\/\/doi.org\/10.1002\/net.3230160109","archive":["Portico"],"relation":{},"ISSN":["0028-3045","1097-0037"],"issn-type":[{"value":"0028-3045","type":"print"},{"value":"1097-0037","type":"electronic"}],"subject":[],"published":{"date-parts":[[1986,3]]}}}