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In extremal graph theory, the forbidden subgraph problem is the following problem: given a graph , find the maximal number of edges (,) an -vertex graph can have such that it does not have a subgraph isomorphic to .
Its authors have divided Elementary Number Theory, Group Theory and Ramanujan Graphs into four chapters. The first of these provides background in graph theory, including material on the girth of graphs (the length of the shortest cycle), on graph coloring, and on the use of the probabilistic method to prove the existence of graphs for which both the girth and the number of colors needed are ...
In spectral graph theory, the Alon–Boppana bound provides a lower bound on the second-largest eigenvalue of the adjacency matrix of a -regular graph, [1] meaning a graph in which every vertex has degree .
Analogously to the classical Fourier transform, the eigenvalues represent frequencies and eigenvectors form what is known as a graph Fourier basis. The Graph Fourier transform is important in spectral graph theory. It is widely applied in the recent study of graph structured learning algorithms, such as the widely employed convolutional networks.
The weighted graph Laplacian: () is a well-studied operator in the graph setting. Mimicking the relationship div ( ∇ f ) = Δ f {\displaystyle \operatorname {div} (\nabla f)=\Delta f} of the Laplace operator in the continuum setting, the weighted graph Laplacian can be derived for any vertex x i ∈ V {\displaystyle x_{i}\in V} as:
In the mathematical field of spectral graph theory, a Ramanujan graph is a regular graph whose spectral gap is almost as large as possible (see extremal graph theory). Such graphs are excellent spectral expanders .
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
In probability theory, the arcsine distribution is the probability distribution whose cumulative distribution function involves the arcsine and the square root: = = +