Graph families | Regular graphs | Algebraic graph theory
In the mathematical field of graph theory, a graph G is symmetric (or arc-transitive) if, given any two pairs of adjacent vertices u1—v1 and u2—v2 of G, there is an automorphism such that and In other words, a graph is symmetric if its automorphism group acts transitively on ordered pairs of adjacent vertices (that is, upon edges considered as having a direction). Such a graph is sometimes also called 1-arc-transitive or flag-transitive. By definition (ignoring u1 and u2), a symmetric graph without isolated vertices must also be vertex-transitive. Since the definition above maps one edge to another, a symmetric graph must also be edge-transitive. However, an edge-transitive graph need not be symmetric, since a—b might map to c—d, but not to d—c. Star graphs are a simple example of being edge-transitive without being vertex-transitive or symmetric. As a further example, semi-symmetric graphs are edge-transitive and regular, but not vertex-transitive. Every connected symmetric graph must thus be both vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree. However, for even degree, there exist connected graphs which are vertex-transitive and edge-transitive, but not symmetric. Such graphs are called half-transitive. The smallest connected half-transitive graph is Holt's graph, with degree 4 and 27 vertices. Confusingly, some authors use the term "symmetric graph" to mean a graph which is vertex-transitive and edge-transitive, rather than an arc-transitive graph. Such a definition would include half-transitive graphs, which are excluded under the definition above. A distance-transitive graph is one where instead of considering pairs of adjacent vertices (i.e. vertices a distance of 1 apart), the definition covers two pairs of vertices, each the same distance apart. Such graphs are automatically symmetric, by definition. A t-arc is defined to be a sequence of t + 1 vertices, such that any two consecutive vertices in the sequence are adjacent, and with any repeated vertices being more than 2 steps apart. A t-transitive graph is a graph such that the automorphism group acts transitively on t-arcs, but not on (t + 1)-arcs. Since 1-arcs are simply edges, every symmetric graph of degree 3 or more must be t-transitive for some t, and the value of t can be used to further classify symmetric graphs. The cube is 2-transitive, for example. (Wikipedia).
Symmetric matrices - eigenvalues & eigenvectors
Free ebook http://tinyurl.com/EngMathYT A basic introduction to symmetric matrices and their properties, including eigenvalues and eigenvectors. Several examples are presented to illustrate the ideas. Symmetric matrices enjoy interesting applications to quadratic forms.
From playlist Engineering Mathematics
What are Symmetric Digraphs? | Graph Theory
What are symmetric digraphs? We'll be defining this class of digraphs with examples in today's graph theory lesson! Remember a digraph is like a regular graph but the edges have direction. Thus, we can describe them as ordered pairs. In directed graphs we call these directed edges "arcs".
From playlist Graph Theory
Diagonalizing a symmetric matrix. Orthogonal diagonalization. Finding D and P such that A = PDPT. Finding the spectral decomposition of a matrix. Featuring the Spectral Theorem Check out my Symmetric Matrices playlist: https://www.youtube.com/watch?v=MyziVYheXf8&list=PLJb1qAQIrmmD8boOz9a8
From playlist Symmetric Matrices
Graph of x^2 + 6xy + 5y^2 rotating
From playlist 3d graphs
From playlist 3d graphs
Symmetric Groups (Abstract Algebra)
Symmetric groups are some of the most essential types of finite groups. A symmetric group is the group of permutations on a set. The group of permutations on a set of n-elements is denoted S_n. Symmetric groups capture the history of abstract algebra, provide a wide range of examples in
From playlist Abstract Algebra
Graph of x^2 + y^2 + pxy as p varies
From playlist 3d graphs
Graph of x^2 + 6xb + 5b^2 as b varies
From playlist 3d graphs
Anna Wienhard (7/29/22): Graph Embeddings in Symmetric Spaces
Abstract: Learning faithful graph representations has become a fundamental intermediary step in a wide range of machine learning applications. We propose the systematic use of symmetric spaces as embedding targets. We use Finsler metrics integrated in a Riemannian optimization scheme, that
From playlist Applied Geometry for Data Sciences 2022
Luis Serrano / The down operator and expansions of near rectangular k-Schur functions
KAIST Discrete Math Seminar Luis Serrano (Université du Québec à Montréal) / 2012-08-07
From playlist Mathematics videos
Bernd Schulze: Characterizing Minimally Flat Symmetric Hypergraphs
Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d-1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). This theory is closely connected to rigidity theory and other areas of discrete applied geometry,
From playlist HIM Lectures 2015
11. Matrix Spaces; Rank 1; Small World Graphs
MIT 18.06 Linear Algebra, Spring 2005 Instructor: Gilbert Strang View the complete course: http://ocw.mit.edu/18-06S05 YouTube Playlist: https://www.youtube.com/playlist?list=PLE7DDD91010BC51F8 11. Matrix Spaces; Rank 1; Small World Graphs License: Creative Commons BY-NC-SA More informat
From playlist MIT 18.06 Linear Algebra, Spring 2005
Shiping Liu (7/29/22): Signed graphs and Nodal domain theorems for symmetric matrices
Abstract: A signed graph is a graph whose edges are labelled by a signature. It serves as a simple model of discrete vector bundle. We will discuss nodal domain theorems for arbitrary symmetric matrices by exploring the induced signed graph structure. This is an extension of the nodal doma
From playlist Applied Geometry for Data Sciences 2022
Analyze the characteristics of multiple functions
👉 Learn about the characteristics of a function. Given a function, we can determine the characteristics of the function's graph. We can determine the end behavior of the graph of the function (rises or falls left and rises or falls right). We can determine the number of zeros of the functi
From playlist Characteristics of Functions
Factorization-based Sparse Solvers and Preconditions, Lecture 3
Xiaoye Sherry Li's (from Lawrence Berkeley National Laboratory) lecture number three on Factorization-based sparse solves and preconditioners
From playlist Gene Golub SIAM Summer School Videos
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From playlist Proof Writing
Sums of Squares Over k-Subset Hypercubes - Annie Raymond
Computer Science/Discrete Mathematics Seminar I Topic: Sums of Squares Over k-Subset Hypercubes Speaker: Annie Raymond Affiliation: University of Massachusetts, Amherst Date: April 16, 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
From playlist 3d graphs
A WEIRD VECTOR SPACE: Building a Vector Space with Symmetry | Nathan Dalaklis
We'll spend time in this video on a weird vector space that can be built by developing the ideas around symmetry. In the process of building a vector space with symmetry at its core, we'll go through a ton of different ideas across a handful of mathematical fields. Naturally, we will start
From playlist The New CHALKboard