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Encyclopedia > Algebraic graph theory

Algebraic graph theory is a branch of mathematics. A diagram of a graph with 6 vertices and 7 edges. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ...

In one sense, algebraic graph theory studies graphs in connection with linear algebra. In particular it studies the spectrum of the adjacency matrix or Laplace matrix of a graph. This part of algebraic graph theory is also called the spectral graph theory. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In most modern usages of the word spectrum, there is a unifying theme of a variety of possible cases between extremes at either end. ... In mathematics and computer science, the adjacency matrix for a finite graph on n vertices is an n × n matrix in which entry aij is the number of edges from vi to vj in . ... In the mathematical field of graph theory the admittance matrix, Kirchhoff matrix, or Laplacian matrix is a matrix representation of a graph. ... In mathematics, spectral graph theory is the study of properties of a graph in relationship to the eigenvalues and eigenvectors of its adjacency matrix. ...

In the other sense, algebraic graph theory studies graphs in connection to group theory (particularly geometric group theory). In particular, the automorphism group of a graph plays an important role. The focus is placed on various families of symmetric graphs such as: vertex-transitive graphs, edge-transitive graphs, arc-transitive graphs, Cayley graphs, etc. Group theory is that branch of mathematics concerned with the study of groups. ... Geometric group theory and combinatorial group theory are two closely related branches of mathematics, which study infinite discrete groups. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... In mathematics, an edge-transitive graph is a graph G such that, given any two edges e1 and e2 of G, there is some automorphism f : G → G such that f ( e1 ) = e2. ... In mathematics, an arc-transitive graph is a graph G such that, given any two edges e1 = u1v1 and e2 = u2v2 of G, there are two automorphisms f : G â†’ G, g : G â†’ G such that f (e1) = e2, g (e1) = e2 and f (u1) = u2, f (v1) = v2, g (u1... The Cayley graph of the free group on two generators a and b In mathematics, a Cayley graph, named after Arthur Cayley, is a graph that encodes the structure of a group. ...

In mathematics, spectral graph theory is the study of properties of a graph in relationship to the eigenvalues and eigenvectors of its adjacency matrix. ... In Graph theory, The Dulmage-Mendelson decomposition is a method used to create an maximal matching on a bipartite graph. ...

## References

• Biggs, Norman (1993). Algebraic Graph Theory (2nd ed.), Cambridge: Cambridge University Press. ISBN 0-521-45897-8.

Results from FactBites:

 Graph theory - Wikipedia, the free encyclopedia (1209 words) Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs) which can be directed (assigned a direction). Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge. A graph drawing should not be confused with the graph itself (the abstract, non-graphical structure) as there are several ways to structure the graph drawing.
 Algebraic graph theory - Wikipedia, the free encyclopedia (144 words) Algebraic graph theory is a branch of mathematics. In one sense, algebraic graph theory studies graphs in connection with linear algebra. This part of algebraic graph theory is also called the spectral graph theory.
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