PDF | In this paper, we obtain explicit formulae forthe total numberof paths oflengths3 and 4 in a simple graph G. We also determine some formulae... |
Complete parts (a) and (b). Fish Lengths and Weights 6 8 1.49 1.76 Length (in.) Weight (lb) 4 1.26 10 2.08 12 2.45 (a) Find an exponential model forthe data.
Find step-by-step Discrete math solutions and your answer to the following textbook question: Findthenumberof paths oflength n between two different...
(A^n)_{ij} holds thenumberof paths oflength from node i to node j . Let’s see how this proposition works. Consider the adjacency matrix of thegraph above
You could use the following interesting result . Let AA denote the adjacency matrix of your graph. Then, the (i,j)(i,j) entry of the matrix AnA^n will denote the number of walks of length nn from vertex ii to vertex jj.
19 Şubat 2016
7 kişi değerlendirdi
2 yorum
{"2_m7nh":{"state":{"logo":{"href":"//yandex.com.tr","isForeign":true},"form":{"action":"/search/","searchLabel":"Bul","hiddenInputs":[{"name":"lr","value":21206}]},"actions":{"advancedSearch":{"checked":false,"ariaLabel":"Gelişmiş arama"},"moreButtonProps":{"ariaLabel":"Daha fazla"},"cbir":{"name":"cbir","label":"Поиск по картинке","infoUrl":"//yandex.ru/promo/photosearch/index?source=landingdesk","isMacOS":false}},"userSection":{"loggedIn":false,"loginProps":{"loginUrl":"https://passport.yandex.com.tr/auth?retpath=%2F&origin=serp_desktop_auth_new","loginText":"Giriş yap","burgerAriaLabel":"Menü"}},"navigation":{"tabs":[{"id":"www","label":"Arama","url":"//yandex.com.tr","params":{"source":"tabbar"},"target":""},{"id":"images","label":"Görsel","url":"//yandex.com.tr/gorsel/search","params":{"text":"[$query]","noreask":"[$noreask]","nomisspell":"[$nomisspell]","parent-reqid":"[$reqid]","from":"tabbar"}},{"id":"video","label":"Video","url":"//yandex.com.tr/video/search","params":{"text":"[$query]","noreask":"[$noreask]","parent-reqid":"[$reqid]","from":"tabbar"}},{"id":"maps","label":"Haritalar","url":"//yandex.com.tr/harita","params":{"text":"[$query]","source":"serp_navig"}},{"id":"translate","label":"Çeviri","url":"//ceviri.yandex.com.tr","params":{"text":"[$query]","from":"tabbar"}}],"selectedTabId":"www"},"context":{"query":"find the number of different walks of length 3 from b to e for the graph below","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL","noreask":"1","lr":"21206","counters":{"type":"b","data":{"$header":"0_7exiz000","logo":"0_7exiz001","login":"0_7exiz003","usermenu":"0_7exiz004","market-cart":"0_7exiz006","more-button":"0_7exiz007","cbir":"0_7exiz008","$navigation":"0_7exiz009","item":"0_7exiz00i","tab-www":"0_7exiz00b","tab-images":"0_7exiz00d","tab-video":"0_7exiz00f","tab-maps":"0_7exiz00h","tab-translate":"0_7exiz00j"}}},"loggedIn":false,"baobab":{"parentNode":{"context":{"genInfo":{"prefix":"0_7exiw01-0-1"},"ui":"desktop","service":"web","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL","fast":{"name":"header-desktop"}}}}},"type":"header-desktop"},"2_7exi0":{"state":{"answersUrl":"https://math.stackexchange.com/questions/1663534/determine-the-number-of-different-walks-of-length-4#answers","type":"fact","tabsOptions":[{"id":"question","text":"Soru"},{"id":"bestAnswer","text":"En iyi yanıt"},{"id":"moreAnswers","text":"1 yanıt daha","shortText":"Daha fazla"}],"tabContents":{"question":{"construct":[{"type":"paragraph","content":[{"type":"text","content":"Let K3K_3 denote the complete graph having 3 vertices. Determine the number of different walks of length 4 there are in K3K_3 from v to v."}]},{"type":"paragraph","content":[{"type":"text","content":"So I used a used the induction rule to find the maximum number of edges (I got 6) and drew the graph out. Considering these are walks, reverse and overlapping walks count. "}]},{"type":"paragraph","content":[{"type":"text","content":"I know I can do this the long way, and go through each walk individually, but I was wondering if there was maybe a simpler much faster way, given that this is a fairly small example and that questions like these can use ridiculously big numbers."}]}],"footerProps":{"date":1455911800,"numLiked":3,"numCommented":3}},"bestAnswer":{"construct":[{"type":"paragraph","content":[{"type":"text","content":"You could use the following interesting result . Let AA denote the adjacency matrix of your graph. Then, the (i,j)(i,j) entry of the matrix AnA^n will denote the number of walks of length nn from vertex ii to vertex jj."}]}],"footerProps":{"date":1455911069,"numLiked":7,"numCommented":2}},"moreAnswers":[{"construct":[{"type":"paragraph","content":[{"type":"text","content":"Let me try this."}]},{"type":"paragraph","content":[{"type":"text","content":"Consider KnK_n, kk steps."}]},{"type":"paragraph","content":[{"type":"text","content":"Total number of walks start from v1=(n−1)kv_1 = (n-1)^k."}]},{"type":"paragraph","content":[{"type":"text","content":"Let NiN_i denotes the number of walks end at viv_i. Clearly {N1,...,Nn}={N1,N2}\\{N_1,...,N_n\\}=\\{N_1,N_2\\}. Let N1=AkN_1=A_k, N2=BkN_2=B_k. "}]},{"type":"paragraph","content":[{"type":"text","content":"One can see that Ak=(n−1)Bk−1A_k=(n-1) B_{k-1}, and Bk=(n−2)Bk−1+Ak−1=(n−2)Bk−1+(n−1)Bk−2B_k=(n-2) B_{k-1}+A_{k-1}=(n-2) B_{k-1}+(n-1)B_{k-2}."}]},{"type":"paragraph","content":[{"type":"text","content":"If loops are not allow (and you can use vertices and edges for more than once), you have"}]},{"type":"paragraph","content":[{"type":"text","content":"00 steps: 11 walk"}]},{"type":"paragraph","content":[{"type":"text","content":"11 steps: 00 walk"}]},{"type":"paragraph","content":[{"type":"text","content":"22 steps: 22 walks"}]},{"type":"paragraph","content":[{"type":"text","content":"33 steps: 22 walks"}]},{"type":"paragraph","content":[{"type":"text","content":"44 steps: 44 walks."}]},{"type":"paragraph","content":[{"type":"text","content":"Then you can add some loops to the sequence of edges to make it a 44-walk."}]},{"type":"paragraph","content":[{"type":"text","content":"This gives you 1+0+2×(52)+2×(4+14)+41+0+2\\times \\binom{5}{2}+2 \\times \\binom{4+1}{4}+4. Then total number of walks is 1+0+10∗2+2∗4+4=331+0+10*2+2*4+4=33"}]}],"footerProps":{"date":1455918862,"numLiked":1,"numCommented":11}}]},"organic":{"faviconProps":{"src":"https://favicon.yandex.net/favicon/v2/https://math.stackexchange.com","size":"m","pageNumber":0,"index":0},"title":"Determine \u0007[the\u0007] \u0007[number\u0007] \u0007[of\u0007] \u0007[different\u0007] \u0007[walks\u0007] \u0007[of\u0007] \u0007[length\u0007] 4.","pathProps":{"items":[{"url":"https://math.stackexchange.com/questions/1663534/determine-the-number-of-different-walks-of-length-4","text":"math.stackexchange.com"},{"url":"https://math.stackexchange.com/questions/1663534/determine-the-number-of-different-walks-of-length-4","text":"determine-the-number-of-different-walks-of-length-4"}]},"faviconPosition":0,"subtitlePosition":1,"linkProps":{"url":"https://math.stackexchange.com/questions/1663534/determine-the-number-of-different-walks-of-length-4"},"hostTitle":"math.stackexchange.com","description":""},"extralinksProps":{"hidden":true,"directions":["bottom-end"],"items":[{"variant":"reportFeedback","reportFeedback":{"feature":"Блок ответов для разработчиков","title":"Bu yanıtta yanlış olan nedir?","checkBoxLabels":[{"value":"Yanıt sorguya uygun değil"},{"value":"Yanıt hatalı"},{"value":"Yanıttaki bilgiler eksik"},{"value":"Diğer"}]}}]},"baobab":{"parentNode":{"context":{"genInfo":{"prefix":"2_7exiw0g-0-1"},"ui":"desktop","service":"web","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL","fast":{"name":"article"}}}}},"type":"article"},"2_7exi1":{"state":{"linksGroups":[{"type":"geo","links":[{"label":"Columbus","title":"Columbus","url":"//yandex.com.tr/tune/geo","logNode":{"name":"region"},"target":"_self","a11yLabel":"Bölgeniz Columbus","needRetpath":true}]},{"type":"feedback","links":[{"label":"Hata bildir","logNode":{"name":"feedback","attrs":{"behaviour":{"type":"dynamic"}}}}]},{"type":"settings","links":[{"label":"Ayarlar","url":"//yandex.com.tr/tune/search","logNode":{"name":"settings"},"target":"_self","needRetpath":true}]},{"type":"company","links":[{"label":"Hakkımızda","url":"https://company.yandex.com.tr","logNode":{"name":"about"},"target":"_blank"},{"label":"Ticari teklifler","url":"/search/direct?text=find%20the%20number%20of%20different%20walks%20of%20length%203%20from%20b%20to%20e%20for%20the%20graph%20below&filters_docs=direct_cm","logNode":{"name":"commercial"},"target":"_blank"},{"label":"Kariyer","url":"//yandex.com.tr/jobs/?utm_source=serp_turkie&utm_medium=footerlink&utm_content=career","logNode":{"name":"vacancy"},"target":"_blank"},{"label":"Kullanım lisansı","url":"//yandex.com.tr/legal/termsofuse/","logNode":{"name":"license"},"target":"_blank"},{"label":"Gizlilik Politikası","url":"//yandex.com.tr/legal/confidential/","logNode":{"name":"confidential"},"target":"_blank"}],"a11yHidden":true}],"reportFeedbackProps":{"feature":"Подвал","title":"Hata bildir","checkBoxLabels":[]},"isSearchApp":false,"hasExtralinks":true,"limitingSelector":".AliceFabButton","baobab":{"parentNode":{"context":{"genInfo":{"prefix":"2_7exiw0h-0-1"},"ui":"desktop","service":"web","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL"}}}},"type":"serp-footer"},"2_7exi2":{"state":{"conditionLoadList":{"marketCart":false,"highchartsHighstock":false,"advProductGallery":false,"musicPlayer":false,"ecomRates":false,"photoViewer":false,"darkmodeChangeDrawer":true},"reportFeedbackBaseProps":{"initEmail":"","metaFields":{"userAgent":"Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)","userTestids":"186144,1060148,1052824,1051377,1046811,1055712,1042347,1010217,1056593,1059425,1048639,1061619,1054624,1056268,1052527,1044021,1031786,138060,936724,1062357,1060761,1064543,1061176,1058044,1046929,1065119,1036351,1045260,1056359,241534,1057945,1051713,1053974,1051607,1060980,1061450,1064443,1063819,1044528,766163,284409,56261,151171,126331,1046868,287509,1055644,338390,997023,1064465,1048648,1060748,1054352,1051465,1052892","yandexuid":"8216645341720899764","uid":"0","queryText":"find the number of different walks of length 3 from b to e for the graph below","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL","searchSlices":["drawer_search_slice_education"],"userRegionName":"Columbus","userRegionId":"20815"}},"extralinksPopupProps":{"abuseLink":"https://twitter.yandex.com.tr/search/abuse?sign=47ff27e24f99814d53d4e9b8b12d898b&keyno=0"},"isLoggedIn":false,"userMenuDialogProps":{"flag":"skin,feedback","host":"yandex.com.tr","lang":"tr","project":"serp","queryParams":{"utm_source":"search"},"retpath":"https%3A%2F%2Ftwitter.yandex.com.tr%2Fsearch%2F%3Fes_context%3D0oYl8Q____________ARpLMTcxOTc0MjE1OTkyMjYzOS04MjU1MDUxNzIyNDMzMDg5MjM2LWJhbGFuY2VyLWw3bGV2ZWxlci1rdWJyLXlwLWtsZy0zMTItQkFMMgVhc3NvYxikQ5I%26lr%3D21206%26noreask%3D1%26rq%3D1%26serp-reload-from%3Drec_bottom%26text%3Dfind%2Bthe%2Bnumber%2Bof%2Bdifferent%2Bwalks%2Bof%2Blength%2B3%2Bfrom%2Bb%2Bto%2Be%2Bfor%2Bthe%2Bgraph%2Bbelow%26ncrnd%3D19385","tld":"com.tr"},"baobab":{"parentNode":{"context":{"genInfo":{"prefix":"2_7exiw0i-0-1"},"ui":"desktop","service":"web","reqid":"1720899765140357-15998859822872902216-balancer-l7leveler-kubr-yp-sas-88-BAL","fast":{"name":"lazy-load"}}}}},"type":"lazy-load"}}