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Undergraduate
By Shiva Kintali on Oct. 12, 2013 | Updated Jan. 4, 2018

Odd sets with even intersection

Let \(\mathcal{F}\) be a family of subsets of \([n]\) such that, for all \(A, B \in \mathcal{F}\) with \(A \neq B\), we have \(|A \cap B|\) is even and \(|A|\) is odd. Prove that …
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Undergraduate
By Shiva Kintali on Oct. 1, 2013 | Updated Jan. 4, 2018

Graphs, Matrices and Walks

Let \(G\) be a directed graph (possibly with self-loops) with vertices \(v_1, \dots , v_n\). Let \(M\) be the adjacency matrix of \(G\). Prove that \(M_{ij}^k\) (i.e., that \([i][j]^{th}\) entry of …
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