{"id":251652,"date":"2025-07-10T19:15:10","date_gmt":"2025-07-10T19:15:10","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=251652"},"modified":"2025-07-10T19:15:12","modified_gmt":"2025-07-10T19:15:12","slug":"how-many-edges-has-each-of-the-following-graphs","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/10\/how-many-edges-has-each-of-the-following-graphs\/","title":{"rendered":"How many edges has each of the following graphs"},"content":{"rendered":"\n

How many edges has each of the following graphs: (a) K10; (b) K5,7; (c) W8; (d) the Petersen graph<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Let’s analyze each of the graphs and their number of edges:<\/p>\n\n\n\n

(a) K10K_{10}K10\u200b (Complete graph on 10 vertices):<\/h3>\n\n\n\n

In a complete graph KnK_nKn\u200b, every pair of distinct vertices is connected by an edge. The number of edges in KnK_nKn\u200b is given by the combination formula:E(Kn)=(n2)=n(n\u22121)2E(K_n) = \\binom{n}{2} = \\frac{n(n-1)}{2}E(Kn\u200b)=(2n\u200b)=2n(n\u22121)\u200b<\/p>\n\n\n\n

For K10K_{10}K10\u200b, where n=10n = 10n=10:E(K10)=10(10\u22121)2=10\u00d792=45E(K_{10}) = \\frac{10(10-1)}{2} = \\frac{10 \\times 9}{2} = 45E(K10\u200b)=210(10\u22121)\u200b=210\u00d79\u200b=45<\/p>\n\n\n\n

So, K10K_{10}K10\u200b has 45 edges<\/strong>.<\/p>\n\n\n\n

(b) K5,7K_{5,7}K5,7\u200b (Complete bipartite graph with partitions of size 5 and 7):<\/h3>\n\n\n\n

In a complete bipartite graph Km,nK_{m,n}Km,n\u200b, every vertex in the first set (size mmm) is connected to every vertex in the second set (size nnn), but there are no edges within each set. The number of edges is:E(Km,n)=m\u00d7nE(K_{m,n}) = m \\times nE(Km,n\u200b)=m\u00d7n<\/p>\n\n\n\n

For K5,7K_{5,7}K5,7\u200b, where m=5m = 5m=5 and n=7n = 7n=7:E(K5,7)=5\u00d77=35E(K_{5,7}) = 5 \\times 7 = 35E(K5,7\u200b)=5\u00d77=35<\/p>\n\n\n\n

Thus, K5,7K_{5,7}K5,7\u200b has 35 edges<\/strong>.<\/p>\n\n\n\n

(c) W8W_8W8\u200b (Wheel graph with 8 vertices):<\/h3>\n\n\n\n

A wheel graph WnW_nWn\u200b is formed by taking a cycle of n\u22121n-1n\u22121 vertices and adding a central vertex connected to all others. The number of edges in WnW_nWn\u200b is given by:E(Wn)=n+(n\u22123)E(W_n) = n + (n-3)E(Wn\u200b)=n+(n\u22123)<\/p>\n\n\n\n

where nnn is the total number of vertices, and the term n\u22123n-3n\u22123 comes from the connections to the center. For W8W_8W8\u200b, where n=8n = 8n=8:E(W8)=8+(8\u22123)=8+5=13E(W_8) = 8 + (8-3) = 8 + 5 = 13E(W8\u200b)=8+(8\u22123)=8+5=13<\/p>\n\n\n\n

So, W8W_8W8\u200b has 13 edges<\/strong>.<\/p>\n\n\n\n

(d) Petersen graph:<\/h3>\n\n\n\n

The Petersen graph is a well-known graph with 10 vertices and 15 edges. It is often used as an example in graph theory due to its interesting properties (such as being a cubic graph, meaning each vertex has degree 3).<\/p>\n\n\n\n

Thus, the Petersen graph has 15 edges<\/strong>.<\/p>\n\n\n\n

Summary:<\/h3>\n\n\n\n
    \n
  • (a) K10K_{10}K10\u200b<\/strong>: 45 edges<\/li>\n\n\n\n
  • (b) K5,7K_{5,7}K5,7\u200b<\/strong>: 35 edges<\/li>\n\n\n\n
  • (c) W8W_8W8\u200b<\/strong>: 13 edges<\/li>\n\n\n\n
  • (d) Petersen graph<\/strong>: 15 edges<\/li>\n<\/ul>\n\n\n\n
    \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

    How many edges has each of the following graphs: (a) K10; (b) K5,7; (c) W8; (d) the Petersen graph The Correct Answer and Explanation is: Let’s analyze each of the graphs and their number of edges: (a) K10K_{10}K10\u200b (Complete graph on 10 vertices): In a complete graph KnK_nKn\u200b, every pair of distinct vertices is connected […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-251652","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251652","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=251652"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/251652\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=251652"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=251652"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=251652"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}