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Bazı konu başlıkları: Graf (Şekil, ağ, network) nedir, düğüm (node, vertex), kenar (Edge, arc) kavramları Grafların küme teorisi ile...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3482714/7f6d76cf27335bd7dac10f2e38501dba/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/dhZkuQAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"12","reqid":"1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=videoid:10575151389641245791","teaser":[{"list":{"type":"unordered","items":["Bu video, Şadi Evren Şeker tarafından \"Bilgisayar Kavramları\" kanalında sunulan, graf teorisi hakkında kapsamlı bir eğitim içeriğidir. Eğitmen, graf teorisinin temel kavramlarını ve uygulamalarını detaylı şekilde anlatmaktadır.","Video, graf teorisinin tanımı ve temel kavramlarıyla başlayıp, düğümler (node/vertices) ve kenarlar (edge) üzerinden grafın yapısal özelliklerini açıklamaktadır. Ardından Königsberg Köprüleri Problemi gibi tarihsel gelişim, yönlü ve yönsüz graf, ağırlıklı graf, bağlantılı graf kavramları, düğüm dereceleri, yürüyüş, yol ve döngü gibi temel kavramlar ele alınmaktadır. Son olarak, ağaç, düzenli graf, iki parçalı graf, tam graf, düzlemsel graf gibi özel graf türleri ve graf gösterim yöntemleri (küme, matris, kenar listesi, düğüm listesi) anlatılmaktadır.","Video, graf teorisinin gerçek hayattaki uygulamalarını da içermekte olup, sosyal ağlar, bilimsel katkılar, tedarik zinciri, biyoinformatics, internet ve ekolojik ağlar gibi alanlardaki graf modelleri örneklerle gösterilmektedir. Ayrıca en kısa yol problemi, gezgin satıcı problemi gibi önemli graf teorisi problemleri de açıklanmaktadır."]},"endTime":3041,"title":"Graf Teorisi Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Graf Teorisi Tanıtımı","list":{"type":"unordered","items":["Konuşmacı Şadi Evren Şeker, \"Bilgisayar Kavramları\" kanalında graf teorisini anlatacaklarını belirtiyor.","Graf (şekil) kavramı, network veya a teorisi olarak da adlandırılabilir ve arkasında küme teorisi yatar.","Graf teorisi, düğümler (node) ve kenarlar (edge) olarak adlandırılan iki temel bileşenden oluşur."]},"beginTime":1,"endTime":71,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1&ask_summarization=1"},{"index":1,"title":"Graf Teorisinin Gerçek Hayatta Kullanımı","list":{"type":"unordered","items":["Gerçek hayatta hemen hemen her şey graflarla modellemek mümkün olabilir; bir varlık birden fazla varlıkla ilişkiliyse orada bir graf vardır.","Kenarlar genelde ilişkiyi, düğümler ise varlıkları gösterir; örneğin arkadaşlar arasındaki güven ilişkileri bir graf olarak gösterilebilir.","Düğümler genellikle isimlerle (sayılar, harfler, şehir isimleri gibi) etiketlenebilir."]},"beginTime":71,"endTime":149,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=71&ask_summarization=1"},{"index":2,"title":"Graf Teorisinin Uygulamaları","list":{"type":"unordered","items":["Sosyal ağlar (Facebook, Twitter, LinkedIn) arkadaşlık ağı olarak graf teorisine örnek verilebilir.","Farklı çevrelerden tanıdığınız kişiler farklı renklerle (renklendirme) gösterilebilir.","Bilimsel çalışmalar, makale atıfları, kurumlar arası ilişkiler, tedarik zinciri ve protein geçişlilik ağı gibi çeşitli alanlarda graf teorisi kullanılır."]},"beginTime":149,"endTime":293,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=149&ask_summarization=1"},{"index":3,"title":"Graf Teorisinin Tarihi","list":{"type":"unordered","items":["Graf teorisi, çözümlenmesi gereken problemlerden ortaya çıkmıştır.","Bu problemlerin çözümü için geliştirilen yöntemler ve tanımlar graf teorinin temelini oluşturmuştur."]},"beginTime":293,"endTime":314,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=293&ask_summarization=1"},{"index":4,"title":"Konigsberg Köprüleri Problemi","list":{"type":"unordered","items":["Konigsberg'de yedi köprü bulunuyor ve bu köprüler bir adadan ve yarımada arasında akan nehirlerle bağlanıyor.","Euler'in ilk teorisi, bu yedi köprüden sadece bir kere geçerek tümünü dolaşabilme problemine odaklanıyor.","Euler, düğümlerin derecesi kavramını ortaya atarak, bir grafa en fazla iki tek sayıda düğüm olabilir veya tamamı çift sayıda düğüm olabilir durumunda tüm kenarlardan bir kere geçilerek bir yol çizilebileceğini ispatladı."]},"beginTime":321,"endTime":454,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=321&ask_summarization=1"},{"index":5,"title":"Graf Teorisinin Uygulamaları","list":{"type":"unordered","items":["Kirşof'un elektrik devreleri üzerindeki teorisi, döngüler ve geçişleri içeren bir graf teorisi örneğidir.","Harita boyama (map coloring) teorisi, komşu bölgeleri aynı renkle boyamamak şartıyla en az sayıda renk kullanma problemidir.","Taşımacılıkta en kısa yol (shortest path problem), gezgin satıcı problemi (travelling salesman problem) gibi çeşitli problemler graf teorisine dayanmaktadır."]},"beginTime":454,"endTime":608,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=454&ask_summarization=1"},{"index":6,"title":"Graf Teorisinin Temel Tanımları","list":{"type":"unordered","items":["Graf (graph), genellikle büyük G harfiyle gösterilir ve düğümler (vertex veya node) ve kenarlar (edge) olmak üzere iki kümeden oluşur.","Graf teorisinde self-loop (kendine dönen bağlantı) kavramı vardır ve bazı düğümler kendi üzerine bağlantıya sahip olabilir.","Yönlü graf (directed graph) içinde, bir düğümden diğerine giden bağlantılar yönüne göre belirlenir ve bir düğümden kendisine dönülemez."]},"beginTime":608,"endTime":773,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=608&ask_summarization=1"},{"index":7,"title":"Yönlü ve Ağırlıklı Graf Kavramları","list":{"type":"unordered","items":["Yönlü graf (directed graph veya degraf olarak da bilinir) içinde, düğümler arasında hem 5'ten 4'e hem de 4'ten 5'e gidilebiliyor, ancak 3'ten 4'e doğrudan gidilemiyor.","Ağırlıklı graf, kenarların üzerinde ağırlık değerleri bulunur; ağırlık mesafe, maliyet veya bir kişi diğerinin paylaşımını beğenen sayı gibi farklı fonksiyonlar olabilir.","Ağırlık fonksiyonları tamamen isteğe bağlıdır ve reel sayılar veya tam sayılar üzerinden tanımlanabilir."]},"beginTime":774,"endTime":883,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=774&ask_summarization=1"},{"index":8,"title":"Graf Bağlantı Türleri","list":{"type":"unordered","items":["Yönsüz graflar, her iki yönde de bağlantı olan çift yönlü yönlü graf olarak da düşünülebilir.","Bağlı graf, her düğümden diğer her düğüme bir şekilde ulaşılabildiği graf türüdür.","Kuvvetli bağlı (strongly connected) graf, her düğümden diğer her düğüme doğrudan gidilebildiği graf türüdür."]},"beginTime":883,"endTime":969,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=883&ask_summarization=1"},{"index":9,"title":"Graf Teorisi Kavramları","list":{"type":"unordered","items":["Component (bileşen) kavramı, tamamen bağlı olmayan bir graf için kullanılır ve birbirinden kopuk olan her bir ada parçası olarak düşünülebilir.","Connected (bağlı) bir grafta sadece bir bileşen vardır, tamamı tek bir bileşen olarak düşünülebilir.","Derece tanımı, her bir düğümden çıkan kenar sayısı veya bir düğümün kaç tane komşusu olduğu olarak tanımlanır."]},"beginTime":974,"endTime":1056,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=974&ask_summarization=1"},{"index":10,"title":"Yönlü ve Yönsüz Graf Teorisi","list":{"type":"unordered","items":["Yönlü graf teorisinde derece ikiye ayrılır: in-degree (gelen derece) ve out-degree (çıkan derece).","Yönlü graf teorisinde toplam derece, in-degree ve out-degree'nin toplamıdır.","Bir graftaki kenar sayısı, düğümlerin toplam derecelerinin yarısıdır (2 × kenar sayısı = toplam dereceler)."]},"beginTime":1056,"endTime":1195,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1056&ask_summarization=1"},{"index":11,"title":"Tek Dereceli Düğümler","list":{"type":"unordered","items":["Tek sayıda derecesi olan düğüm sayısı çift olmalıdır.","Tek dereceli düğüm en az bir olabilir ve bir düğümün tek dereceye sahip olması durumunda, onu dengeleyen başka bir tek dereceye sahip düğüm de olmalıdır.","Çift sayıda tek dereceye sahip düğüm olabilir, tek sayıda tek dereceye sahip düğüm olamaz."]},"beginTime":1195,"endTime":1263,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1195&ask_summarization=1"},{"index":12,"title":"Yürüyüş ve Yollar","list":{"type":"unordered","items":["Walk (yürüyüş), kenar ve düğümlerin tekrar olabileceği bir yol olarak tanımlanır.","Path, tekrar olmayan bir yürüyüş olarak tanımlanır; aynı düğümden geçmek mümkündür ancak aynı kenardan geçmek yasaktır.","Kapalı yürüyüş (closed walk), başlangıç ve bitiş noktaları aynı olan bir yürüyüştir."]},"beginTime":1263,"endTime":1471,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1263&ask_summarization=1"},{"index":13,"title":"Özel Graf Türleri","list":{"type":"unordered","items":["Boş graf veya nul graf, hiç kenarı olmayan graf olarak tanımlanır.","Tamamen boş graf, hem kenarlar hem de düğümler kümesi boş olan graf olarak tanımlanır.","Ağaç, cycle (döngü) içermeyen bir graf olarak tanımlanır, yani başlangıç noktasına dönen bir yol bulunmaz."]},"beginTime":1471,"endTime":1519,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1471&ask_summarization=1"},{"index":14,"title":"Ağaçlar ve Özel Ağaç Türleri","list":{"type":"unordered","items":["Ağaç, iki düğümü bağlayan tek bir yol olan ve kopuk olmayan bir grafik türüdür.","Özel ağaçlara \"yıldız ağacı\" (star tree) ve \"yol ağacı\" (path tree) gibi isimler verilir.","Bilgisayar bilimlerinde ağaçlar genellikle \"directed acyclic graph\" (DAG) olarak tanımlanır, ancak bazı kaynaklarda directed olma şartı zorunlu değildir."]},"beginTime":1523,"endTime":1650,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1523&ask_summarization=1"},{"index":15,"title":"Düzenli Grafikler","list":{"type":"unordered","items":["Düzenli (regular) grafik, bağlı ve tüm düğümlerinin derecesi aynı olan grafiklerdir.","Regular cycle, bağlı ve tüm düğümlerin derecesi iki olan grafiklerdir."]},"beginTime":1650,"endTime":1704,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1650&ask_summarization=1"},{"index":16,"title":"İki Parçalı Grafikler","list":{"type":"unordered","items":["İki parçalı grafik (bipartite graph), düğümleri iki gruba bölünebilen ve kenarlar sadece farklı gruplardaki düğümleri birleştiren grafiklerdir.","İki parçalı grafikte, bir kenarın uçları aynı grupta olamaz."]},"beginTime":1704,"endTime":1786,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1704&ask_summarization=1"},{"index":17,"title":"Tam Grafikler","list":{"type":"unordered","items":["Tam grafik (complete graph) içindeki her düğüm, diğer tüm düğümlere doğrudan bağlıdır.","Tam grafikte düğüm sayısı n ise kenar sayısı n×(n-1)/2 olur.","Tam iki parçalı grafikte, bir gruptaki her düğüm diğer gruptaki tüm düğümlere bağlıdır."]},"beginTime":1786,"endTime":1922,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1786&ask_summarization=1"},{"index":18,"title":"Düzlemsel Grafikler","list":{"type":"unordered","items":["Düzlemsel grafikler (planar graph), düğümleri ve kenarları düzlemde çizilebilen grafiklerdir."]},"beginTime":1922,"endTime":1931,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1922&ask_summarization=1"},{"index":19,"title":"Graf Teorisi Kavramları","list":{"type":"unordered","items":["Dört düğüme sahip bir grafın maksimum altı kenar içerebilir, yedinci bir kenar hiçbir şekilde çizilemez çünkü ya daha öncekilerden birinin tekrarı olacaktır ya da birileriyle kesişecektir.","Planner graf ile kastedilen kesişen düğüm kenar olmaması durumdur.","Alt graf (subgraph) bir grafın düğümler kümesinin ve kenarlar kümesinin alt kümesi olarak tanımlanır."]},"beginTime":1935,"endTime":2059,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=1935&ask_summarization=1"},{"index":20,"title":"Click ve Spanning Subgraph Kavramları","list":{"type":"unordered","items":["Click (klik), bir grafın içinde bulunan en fazla kenar sayısına sahip veya en fazla düğüm sayısına sahip, doğrudan birbirine bağlı tam alt graf (complete subgraph) olarak tanımlanır.","Spanning subgraph (kapsayan alt graf), bir grafın bütün düğümlerini içeren ancak tüm kenarları içermesi gerekmayan bir alt graftır.","Tarama ağacı (spanning tree), bütün düğümleri ve kenarları içeren, ağaç tanımına uyan ve cycle içermeyen bir alt graftır."]},"beginTime":2059,"endTime":2260,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=2059&ask_summarization=1"},{"index":21,"title":"İzomorfizm Kavramı","list":{"type":"unordered","items":["İzomorfizm, birebir (one-to-one) ve örtün (onto) olan bir fonksiyonu gerektiren, iki graf arasında komşulukları koruyan bir eşleme ilişkisidir.","İzomorfik graflar arasında toplam kenar sayısı, toplam düğüm sayısı ve düğüm dereceleri aynıdır.","İzomorfik graflar, çizimlerinde düğümlerin yerlerini değiştirerek birbirine dönüştürülebilir ve bu kavram veri güvenliğinde de kullanılmaktadır."]},"beginTime":2260,"endTime":2445,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=2260&ask_summarization=1"},{"index":22,"title":"Grafların Farklı Gösterimleri","list":{"type":"unordered","items":["Graflar genellikle düğümleri ve kenarları içeren iki kümenin birleşimi olarak tanımlanır, ancak farklı gösterimler de mümkündür.","Graflar matris olarak da gösterilebilir; bu matrisler incidence matris (durum matrisi) ve adjacency matris (komşuluk matrisi) olarak adlandırılır.","Incidence matris kenar ve düğümleri birlikte tutarken, adjacency matris sadece kenarları tutar; yönsüz graf'larda köşegen üzerinden simetriktir, yönlü graf'larda ise olmayabilir."]},"beginTime":2448,"endTime":2632,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=2448&ask_summarization=1"},{"index":23,"title":"Grafların Diğer Gösterimleri","list":{"type":"unordered","items":["Graflar kenar listesi (edge list) veya düğüm listesi (node list) olarak da gösterilebilir.","Edge list'te her kenar bağımlı düğümleriyle birlikte listelenir, node list'te ise her düğümün bağlandığı düğümleri gösterilir.","Ağırlıklı graflarda edge list'te ilave olarak ağırlık bilgisi de eklenir."]},"beginTime":2632,"endTime":2754,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=2632&ask_summarization=1"},{"index":24,"title":"Topolojik Mesafe ve Mesafe Matrisi","list":{"type":"unordered","items":["Topolojik mesafe (topological distance), her düğümden geçmenin maliyetinin bir olduğu durumlarda hesaplanır.","Mesafe matrisi, bir düğümden diğer düğümlere ulaşma maliyetlerini gösterir ve Dijkstra gibi kısa yol algoritmalarıyla hesaplanabilir.","Bu matris, eski haritalarda yer alan şehirler arası mesafe tablolarına benzer şekilde kullanılır."]},"beginTime":2754,"endTime":2892,"href":"/video/preview/10575151389641245791?parent-reqid=1769787930241257-9005928562935062751-balancer-l7leveler-kubr-yp-klg-122-BAL&text=G-Rapp+The+General+-+Topic&t=2754&ask_summarization=1"},{"index":25,"title":"Graf Teorisi Kavramları","list":{"type":"unordered","items":["Bu video, graf teorisi için temel kavramları ve gösterim yöntemlerini içermektedir.","Graf teorisi alanında daha fazla kavram ve problemler bulunmaktadır, örneğin maksimum akış-minimum kesit problemi, minimum ağaç problemi gibi.","Bu videoda graf teorisi, kavramlar, gösterim ve kullanım yöntemleri 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Her bir konu için örnekler verilerek, tablo ve grafiklerden hangi bilgilerin çıkarılabileceği, hangi bilgilerin çıkarılamayacağı açıklanmaktadır. Ayrıca görsel yorumlama ve kroki okuma konularında da örnekler sunulmaktadır. Video, bir etkinlik çalışması ve soru çözüm videosu ile devam edecektir."]},"endTime":437,"title":"8. 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