Discrete Mathematics in Statistical Physics: Introductory by Martin Loebl

By Martin Loebl

The ebook first describes connections among a few uncomplicated difficulties and technics of combinatorics and statistical physics. The discrete arithmetic and physics terminology are with regards to one another. utilizing the proven connections, a few intriguing actions in a single box are proven from a standpoint of the opposite box. the aim of the e-book is to stress those interactions as a powerful and profitable device. actually, this angle has been a powerful development in either study groups lately.
It additionally obviously ends up in many open difficulties, a few of which appear to be simple. confidently, this ebook may also help making those interesting difficulties appealing to complex scholars and researchers.

uncomplicated ideas - advent to Graph idea - timber and electric networks – Matroids - Geometric representations of graphs - video game of dualities - The zeta functionality and graph polynomials – Knots - second Ising and dimer models

- complicated Graduate scholars in arithmetic, Physics and laptop Sciences
- Researchers

Prof. Dr. Martin Loebl, Dept. of arithmetic, Charles college, Prague

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Discrete Mathematics in Statistical Physics: Introductory Lectures

The publication first describes connections among a few simple difficulties and technics of combinatorics and statistical physics. The discrete arithmetic and physics terminology are with regards to one another. utilizing the validated connections, a few intriguing actions in a single box are proven from a standpoint of the opposite box.

Additional info for Discrete Mathematics in Statistical Physics: Introductory Lectures

Sample text

If there is an edge in G between two vertices with an even distance to a (possibly different) root then there is an augmenting path, or a flower and thus a blossom of a matching of the same size as M . We can contract the blossom and continue with the smaller graph. 9. e. any graph that can be drawn in the plane without edge-crossings, may be assigned one of four colors so that no edge has its vertices of the same color. This theorem still has only a computer-assisted proof. 4). 10. Let G = (V, E) be a graph.

We say that Y ⊂ X is a system of distinct representatives for A if the elements of Y may be numbered so that the i-th element belongs to Ai , i = 1, · · · , m. 6. A family A = {A1 , · · · , Am } of subsets of a set X has a system of distinct representatives if and only if | Ai | ≥ |F | i∈F for every F ⊂ {1, · · · , m}. Given a general matrix A where the rows are indexed by a set V1 and the columns are indexed by a set V2 , we can form its bipartite graph (V1 ∪ V2 , E) where ij ∈ E if and only if Aij = 1.

A graph has a matching covering all but at most d vertices if and only if for each S ⊂ V , the number of components of G − S of an odd cardinality is at most |S| + d. A graph is factor-critical if it has a perfect matching after deletion of an arbitrary vertex. 8 proves more: it demonstrates a useful graph decomposition, the Edmonds-Gallai decomposition. 6. 10. Let G = (V, E) be a graph. Let C be the set of the vertices of G not covered by at least one maximum matching. Let S ⊂ V − C be the set of the neighbours of the vertices of C, and let D be the set of the remaining vertices.

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