**Problem 1:** Find the maximum possible number of edges in a bipartite graph of order \( n \) for any integer \( n > 1 \).### Explanation:A bipartite graph is a type of graph where the vertex set can be divided into two disjoint subsets such that no two graph vertices within the same subset are adjacent. The maximum number of edges in a bipartite graph occurs when the graph is a complete bipartite graph, denoted as \( K_{m,n} \). The number of edges \( E \) in a complete bipartite graph \( K_{m,n} \) is given by:\[ E = m \times n \]For a bipartite graph of order \( n \) (total \( n \) vertices), the subsets can have sizes \( m \) and \( n-m \), where \( 0 < m < n \). The maximum number of edges is obtained when:\[ E = m \times (n-m) \]To maximize \( E \), \( m \) and \( (n-m) \) should be as equal as possible. Thus, for maximum edges, the two parts should be approximately equal in size, leading to:\[ E_{\text{max}} = \left\lfloor \frac{n^2}{4} \right\rfloor \]This formula gives the maximum number of edges for a bipartite graph with \( n \) vertices.

The bipartite graph on n vertices with the maximum edges is clearly a complete bipartite graph,…

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Problem 1: find the maximum possible number of edges in ipartite graph of order n for any integer n > 1.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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