{"id":61,"date":"2022-02-20T14:50:36","date_gmt":"2022-02-20T14:50:36","guid":{"rendered":"http:\/\/localhost\/kpv\/?p=61"},"modified":"2025-02-14T17:37:22","modified_gmt":"2025-02-14T17:37:22","slug":"line-graphs","status":"publish","type":"post","link":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/line-graphs\/","title":{"rendered":"Line graphs"},"content":{"rendered":"\n<p>In graph theory, the <strong>line graph<\/strong> of a given graph G, denoted as <strong>L(G)<\/strong>, is a graph that represents the <strong>adjacency relationships between the edges of G<\/strong>. Specifically, <strong>L(G)<\/strong> contains one vertex for each edge in G, and two vertices in <strong>L(G)<\/strong> are considered adjacent if and only if their corresponding edges in G share a common vertex*.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>Construction of L(G) from G:<\/strong><\/h3>\n\n\n\n<p>The transformation from <strong>G<\/strong> to <strong>L(G)<\/strong> follows a systematic procedure to preserve the structural relationships between edges while redefining them as vertices. The steps for constructing <strong>L(G)<\/strong> are outlined below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"530\" height=\"448\" src=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph1-1.png\" alt=\"\" class=\"wp-image-228\" style=\"width:265px;height:224px\" srcset=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph1-1.png 530w, https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph1-1-300x254.png 300w\" sizes=\"auto, (max-width: 530px) 100vw, 530px\" \/><\/figure>\n\n\n\n<p>For each edge present in <strong>G<\/strong>, a corresponding vertex is introduced in <strong>L(G)<\/strong>, ensuring that every edge in the original graph is represented as a distinct node in the transformed line graph.<br><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"530\" height=\"386\" src=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph2-1.png\" alt=\"\" class=\"wp-image-229\" style=\"width:265px;height:193px\" srcset=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph2-1.png 530w, https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-content\/uploads\/2022\/02\/lineGraph2-1-300x218.png 300w\" sizes=\"auto, (max-width: 530px) 100vw, 530px\" \/><\/figure>\n\n\n\n<p id=\"j-l-gross-and-j-yellen-graph-theory-and-its-applications-2005\" style=\"font-size:10px\">*J. L. Gross and J. Yellen, Graph Theory and Its Applications. 2005.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In graph theory, the line graph of a given graph G, denoted as L(G), is a graph that represents the adjacency relationships between the edges&#8230;<\/p>\n<div class=\"more-link-wrapper\"><a class=\"more-link\" href=\"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/line-graphs\/\">Continue reading<span class=\"screen-reader-text\">Line graphs<\/span><\/a><\/div>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-61","post","type-post","status-publish","format-standard","hentry","category-linegraph","entry"],"_links":{"self":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/61","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/comments?post=61"}],"version-history":[{"count":6,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/61\/revisions"}],"predecessor-version":[{"id":347,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/posts\/61\/revisions\/347"}],"wp:attachment":[{"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/media?parent=61"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/categories?post=61"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/dalmolingroup.imd.ufrn.br\/kpv\/wp-json\/wp\/v2\/tags?post=61"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}