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Graduate
By
Shiva Kintali
on June 6, 2012 | Updated Dec. 6, 2017
Algebraic dual of graphs
Let \(G\) be a connected graph. An algebraic dual of \(G\) is a graph \(G'\) such that \(G\) and \(G'\) have the same set of edges, any cycle of \(G\) is a cut of \(G'\), and any cut of \(G\) is a cyc…
Mathematics
Graph Theory
matroids
0
Undergraduate
By
Shiva Kintali
on Aug. 22, 2013 | Updated Jan. 4, 2018
Graphs and Fermat's Little Theorem
Given a prime number \(n\), let \(\mathbb{Z}_n\), denote the set of congruence classes of integers modulo \(n\). Let \(a\) be a natural number having no common prime factors with \(n\); multiplication…
Mathematics
Graph Theory
Number Theory
digraphs
fermats little theorem
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Undergraduate
By
Shiva Kintali
on July 5, 2012 | Updated Dec. 6, 2017
Cycle in k-connected graphs
Prove that every \(k\)-connected graph (\(k > 1\)) on at least \(2k\) vertices has a cycle of length at least \(2k\).
Mathematics
Graph Theory
connectivity
0
Undergraduate
By
Shiva Kintali
on Nov. 12, 2012 | Updated Dec. 6, 2017
Alice, Bob and children
Alice and Bob decide to have children until either they have their first girl or they have \(k \geq 1\) children. Assume that each child is a boy or girl independently with probability 1/2, and that …
Mathematics
Probability
expectation
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Large min-degree implies perfect matching
Let \(G\) be a bipartite graph with partitions \(X\) and \(Y\) such that \(|X|=|Y|=n\). The degree of each vertex in \(G\) is at least \(n/2\). Prove that \(G\) has a perfect matching.
Mathematics
Graph Theory
matching
0
Undergraduate
By
Shiva Kintali
on May 19, 2013 | Updated Dec. 6, 2017
Connectivity of dual graph
Prove that if \(G\) is a simple 3-connected planar graph with at least 4 vertices, then the dual of \(G\) is also a simple 3-connected planar graph.
Mathematics
Graph Theory
planar graphs
0
High School
By
Shiva Kintali
on Jan. 5, 2018
Evaluate Riemann Zeta function at all positive even integers
Reimann zeta function is the following \(\zeta(s) = \displaystyle\sum_{n=1}^{\infty}{\frac{1}{n^s}}\) Euler proved the following …
Mathematics
Mathematical Analysis
infinite series
riemann zeta function
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