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Many different branches of mathematics are combined together. For example, algebra and topology combine to give Algebraic Topology; algebra and geometry give rise to Algebraic Geometry, etc.

What is the name of the branch which combines Discrete Mathematics and Linear Algebra, if such a branch exists? If not, which subfields of discrete mathematics make significant use of linear algebra and do they have specific names?

Théophile
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adam
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4 Answers4

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Spectral graph theory is a subfield in which linear algebra is used heavily to study discrete mathematics. Wikipedia:

"In mathematics, spectral graph theory is the study of properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated to the graph, such as its adjacency matrix or Laplacian matrix."

Community
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user1551
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  • I would say that spectral graph theory falls under algebraic graph theory, but it's worth pointing out in its own right. – J W Mar 28 '15 at 09:57
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Maybe you're looking for algebraic combinatorics?

Among other things, it deals with matroids, which is basically the first thing that popped into my head when I read your question.

J W
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Benjamin Lindqvist
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Discrete mathematics is simply too diverse for there to be a single named branch of mathematics that combines it with linear algebra. (See the Wikipedia article to get an impression of discrete mathematics' breadth.)

That said, when it comes to subfields, graph theory is a major subarea under the umbrella of discrete mathematics and algebraic graph theory draws heavily on both linear algebra and group theory. (See, for instance, Algebraic Graph Theory by Godsil and Royle.)

As mentioned in luegofuego's answer, the subfield of algebraic combinatorics uses concepts and techniques from abstract algebra and linear algebra. For matroid theory in particular, you might want to take a look at Matroids: A Geometric Introduction by Gordon and McNulty.

If you extend the umbrella far enough, you can include linear programming/optimization, which certainly uses linear algebra, but does not include algebraic in the name of the field, although the adjective linear does appear.

Community
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J W
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I believe what you are looking for is Discrete Geometry. I say so because;

  1. At the beginning of Discrete Geometry, you define notions like "affine combination" and "convex hull" which are inherited from Linear Algebra. For this you can check any Discrete Geometry textbook, and if you need recommendations I would definitely suggest Lectures on Discrete Geometry by Jiří Matoušek. It is a dense book but easily readable if you are familiar with things like compactness.
  2. As you have it in the name, Discrete Geometry deals with combinatorial properties of objects like convex sets, vertices, hyperplanes and edges.

I hope this helps!

CeyhunElmacioglu
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