Day 07Part 05- Discrete mathematics Graph Theory-Concept of Complete bipartite graph<\/p>\n
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A bipartite graph<\/strong> is a type of graph G = (V, E)<\/strong> where:<\/p>\n A complete bipartite graph<\/strong>, denoted as K\u2098,\u2099<\/strong>, is a special type of bipartite graph where: Consider K\u2083,\u2084<\/strong>, where:<\/p>\n Total edges = 3 \u00d7 4 = 12<\/strong><\/p>\n No Odd-Length Cycles<\/strong> \u2013 A bipartite graph cannot have odd-length cycles<\/strong>. Job Assignment Problem<\/strong> \u2013 Connecting workers (U) to tasks (V). A complete bipartite graph K\u2098,\u2099<\/strong> is a fundamental structure in graph theory, representing maximum connections<\/strong> between two independent sets. Its applications in real-world problems make it an essential concept in discrete mathematics, networking, and optimization<\/strong>.<\/p>\n Would you like an example problem or visualization for better understanding?<\/p>\n Here\u2019s a clear explanation for Discrete Mathematics \u2013 Day 07 Part 05: Complete Bipartite Graph<\/strong> under Graph Theory<\/strong>:<\/p>\n A bipartite graph<\/strong> is a graph where:<\/p>\n The vertices can be divided<\/strong> into two disjoint sets: U<\/strong> and V<\/strong><\/p>\n<\/li>\n Every edge<\/strong> connects a vertex in U<\/strong> to a vertex in V<\/strong><\/p>\n<\/li>\n No edge exists within<\/strong> the same set<\/p>\n<\/li>\n<\/ul>\n A complete bipartite graph<\/strong> is a special type of bipartite graph where:<\/p>\n Every vertex in set U<\/strong> is connected to every vertex in set V<\/strong><\/p>\n<\/li>\n<\/ul>\n It is denoted as:<\/p>\n Where:<\/p>\n m = number of vertices in set U<\/strong><\/p>\n<\/li>\n n = number of vertices in set V<\/strong><\/p>\n<\/li>\n<\/ul>\n Total vertices = m + n<\/strong><\/p>\n<\/li>\n Total edges = m \u00d7 n<\/strong><\/p>\n<\/li>\n It has no cycles of odd length<\/strong><\/p>\n<\/li>\n Example:<\/p>\n K<sub>2,3<\/sub><\/strong> has 2 vertices in one set and 3 in the other \u2192 total 6 edges<\/p>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n
\u00a0What is a Complete Bipartite Graph?<\/strong><\/h3>\n
\u00a0Every vertex in set U (m vertices)<\/strong> is connected to every vertex in set V (n vertices)<\/strong>.
\u00a0There are m \u00d7 n edges<\/strong> (each vertex in U connects to every vertex in V).
\u00a0It is maximally connected<\/strong> for a bipartite structure.<\/p>\n\u00a0Example of a Complete Bipartite Graph (K\u2083,\u2084)<\/strong><\/h3>\n
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Graph Representation:<\/strong><\/h4>\n
A<\/span> -- 1<\/span> B<\/span> -- 1<\/span> C -- 1<\/span>
\nA<\/span> -- 2<\/span> B<\/span> -- 2<\/span> C -- 2<\/span>
\nA<\/span> -- 3<\/span> B<\/span> -- 3<\/span> C -- 3<\/span>
\nA<\/span> -- 4<\/span> B<\/span> -- 4<\/span> C -- 4<\/span>
\n<\/code><\/div>\n<\/div>\n\u00a0Properties of a Complete Bipartite Graph<\/strong><\/h3>\n
Chromatic Number (\u03c7) = 2<\/strong> \u2013 Since it\u2019s bipartite, it only requires two colors<\/strong> for proper coloring.
Degree of Each Vertex<\/strong><\/p>\n\n
\u2714 Planarity<\/strong><\/li>\n\u00a0Real-World Applications of Complete Bipartite Graphs<\/strong><\/h3>\n
Network Design<\/strong> \u2013 Connecting data centers and servers<\/strong> efficiently.
Social Network Analysis<\/strong> \u2013 Representing relationships between two distinct groups (e.g., authors and research papers).
Computer Science & AI<\/strong> \u2013 Used in matching algorithms and recommendation systems.<\/p>\n\u00a0Conclusion<\/strong><\/h3>\n
Day 07Part 05- Discrete mathematics Graph Theory-Concept of Complete bipartite graph<\/a><\/h3>\n
GRAPH THEORY WITH APPLICATIONS<\/a><\/h3>\n
Bipartite Graphs Discrete Mathematics<\/a><\/h3>\n
Lecture 17: Graph Theory Bipartite Graphs<\/a><\/h3>\n
\n\ud83d\udcca Complete Bipartite Graph \u2013 Concept<\/strong><\/h2>\n
\ud83d\udd39 What is a Bipartite Graph?<\/strong><\/h3>\n
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\n\ud83d\udd38 What is a Complete Bipartite Graph?<\/strong><\/h3>\n
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\ud83d\udc49 K<sub>m,n<\/sub><\/strong><\/h3>\n
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\n\ud83e\udde0 Properties of Complete Bipartite Graph K<sub>m,n<\/sub>:<\/strong><\/h3>\n
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\n\ud83d\udd0d Visual Example (K\u2082,\u2083):<\/strong><\/h3>\n