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lucatrevisan.wordpress.com | ||
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www.jeremykun.com
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| | | | | For fixed integers $ r > 0$, and odd $ g$, a Moore graph is an $ r$-regular graph of girth $ g$ which has the minimum number of vertices $ n$ among all such graphs with the same regularity and girth. (Recall, A the girth of a graph is the length of its shortest cycle, and it's regular if all its vertices have the same degree) Problem (Hoffman-Singleton): Find a useful constraint on the relationship between $ n$ and $ r$ for Moore graphs of girth $ 5$ and degree $ r$. | |
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nhigham.com
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| | | | | The spectral radius $latex \rho(A)$ of a square matrix $latex A\in\mathbb{C}^{n\times n}$ is the largest absolute value of any eigenvalue of $LATEX A$: $latex \notag \rho(A) = \max\{\, |\lambda|: \lambda~ \mbox{is an eigenvalue of}~ A\,\}. $ For Hermitian matrices (or more generally normal matrices, those satisfying $LATEX AA^* = A^*A$) the spectral radius is just... | |
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gowers.wordpress.com
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| | | | | Here is a simple but important fact about bipartite graphs. Let $latex G$ be a bipartite graph with (finite) vertex sets $latex X$ and $latex Y$ and edge density $latex \alpha$ (meaning that the number of edges is $latex \alpha |X||Y|$). Now choose $latex (x_1,x_2)$ uniformly at random from $latex X^2$ and $latex (y_1,y_2)$ uniformly | |
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www.jeremykun.com
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| | | This proof assumes knowledge of complex analysis, specifically the notions of analytic functions and Liouville's Theorem (which we will state below). The fundamental theorem of algebra has quite a few number of proofs (enough to fill a book!). In fact, it seems a new tool in mathematics can prove its worth by being able to prove the fundamental theorem in a different way. This series of proofs of the fundamental theorem also highlights how in mathematics there are many many ways to prove a single theorem... | ||
