Algebraic Graph Theory Morphisms Monoids and Matrices 1st Edition by Ulrich Knauer ISBN 3110254085 9783110254082

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ISBN 10: 3110254085
ISBN 13: 9783110254082
Author: Ulrich Knauer

Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures – like roads, computers, telephones – instances of abstract data structures – like lists, stacks, trees – and functional or object oriented programming. In turn, graphs are models for mathematical objects, like categories and functors.

This highly self-contained book about algebraic graph theory is written with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. It ends with a challenging chapter on the topological question of embeddability of Cayley graphs on surfaces.

Algebraic Graph Theory Morphisms Monoids and Matrices 1st Edition Table of contents:

1 Directed and undirected graphs

1.1 Formal description of graphs

1.2 Connectedness and equivalence relations

1.3 Some special graphs

1.4 Homomorphisms

1.5 Half-, locally, quasi-strong and metric homomorphisms

1.6 The factor graph, congruences, and the Homomorphism Theorem

Factor graphs

The Homomorphism Theorem

1.7 The endomorphism type of a graph

1.8 Comments

2 Graphs and matrices

2.1 Adjacency matrix

Isomorphic graphs and the adjacency matrix

Components and the adjacency matrix

Adjacency list

2.2 Incidence matrix

2.3 Distances in graphs

The adjacency matrix and paths

The adjacency matrix, the distance matrix and circuits

2.4 Endomorphisms and commuting graphs

2.5 The characteristic polynomial and eigenvalues

2.6 Circulant graphs

2.7 Eigenvalues and the combinatorial structure

Cospectral graphs

Eigenvalues, diameter and regularity

Automorphisms and eigenvalues

2.8 Comments

3 Categories and functors

3.1 Categories

Categories with sets and mappings, I

Constructs, and small and large categories

Special objects and morphisms

Categories with sets and mappings, II

Categories with graphs

Other categories

3.2 Products & Co

Coproducts

Products

Tensor products

Categories with sets and mappings, III

3.3 Functors

Covariant and contravariant functors

Composition of functors

Special functors – examples

Mor functors

Properties of functors

3.4 Comments

4 Binary graph operations

4.1 Unions

The union

The join

The edge sum

4.2 Products

The cross product

The coamalgamated product

The disjunction of graphs

4.3 Tensor products and the product in EGra

The box product

The boxcross product

The complete product

Synopsis of the results

Product constructions as functors in one variable

4.4 Lexicographic products and the corona

Lexicographic products

The corona

4.5 Algebraic properties

4.6 Mor constructions

Diamond products

Left inverses for tensor functors

Power products

Left inverses to product functors

4.7 Comments

5 Line graph and other unary graph operations

5.1 Complements, opposite graphs and geometric duals

5.2 The line graph

Determinability of G by LG

5.3 Spectra of line graphs

Which graphs are line graphs?

5.4 The total graph

5.5 The tree graph

5.6 Comments

6 Graphs and vector spaces

6.1 Vertex space and edge space

The boundary & Co

Matrix representation

6.2 Cycle spaces, bases & Co

The cycle space

The cocycle space

Orthogonality

The boundary operator & Co

6.3 Application: MacLane’s planarity criterion

6.4 Homology of graphs

Exact sequences of vector spaces

Chain complexes and homology groups of graphs

6.5 Application: number of spanning trees

6.6 Application: electrical networks

6.7 Application: squared rectangles

6.8 Application: shortest (longest) paths

6.9 Comments

7 Graphs, groups and monoids

7.1 Groups of a graph

Edge group

7.2 Asymmetric graphs and rigid graphs

7.3 Cayley graphs

7.4 Frucht-type results

Frucht’s theorem and its generalization for monoids

7.5 Graph-theoretic requirements

Smallest graphs for given groups

Additional properties of group-realizing graphs

7.6 Transformation monoids and permutation groups

7.7 Actions on graphs

Fixed-point-free actions on graphs

Transitive actions on graphs

Regular actions

7.8 Comments

8 The characteristic polynomial of graphs

8.1 Eigenvectors of symmetric matrices

Eigenvalues and connectedness

Regular graphs and eigenvalues

8.2 Interpretation of the coefficients of chapo(G)

Interpretation of the coefficients for undirected graphs

8.3 Spectra of trees

Recursion formula for trees

8.4 The spectral radius of undirected graphs

Subgraphs

Upper bounds

Lower bounds

8.5 Spectral determinability

Spectral uniqueness of Kn and Kp;q

8.6 Eigenvalues and group actions

Groups, orbits and eigenvalues

8.7 Transitive graphs and eigenvalues

Derogatory graphs

Graphs with Abelian groups

8.8 Comments

9 Graphs and monoids

9.1 Semigroups

9.2 End-regular bipartite graphs

Regular endomorphisms and retracts

End-regular and End-orthodox connected bipartite graphs

9.3 Locally strong endomorphisms of paths

Undirected paths

Directed paths

Algebraic properties of LEnd

9.4 Wreath product of monoids over an act

9.5 Structure of the strong monoid

The canonical strong decomposition of G

Decomposition of SEnd

A generalized wreath product with a small category

Cardinality of SEnd(G)

9.6 Some algebraic properties of SEnd

Regularity and more for TA

Regularity and more for SEnd(G)

9.7 Comments

10 Compositions, unretractivities and monoids

10.1 Lexicographic products

10.2 Unretractivities and lexicographic products

10.3 Monoids and lexicographic products

10.4 The union and the join

The sum of monoids

The sum of endomorphism monoids

Unretractivities

10.5 The box product and the cross product

Unretractivities

The product of endomorphism monoids

10.6 Comments

11 Cayley graphs of semigroups

11.1 The Cay functor

Reflection and preservation of morphisms

Does Cay produce strong homomorphisms?

11.2 Products and equalizers

Categorical products

Equalizers

Other product constructions

11.3 Cayley graphs of right and left groups

11.4 Cayley graphs of strong semilattices of semigroups

11.5 Application: strong semilattices of (right or left) groups

11.6 Comments

12 Vertex transitive Cayley graphs

12.1 Aut-vertex transitivity

12.2 Application to strong semilattices of right groups

ColAut(S,C)-vertex transitivity

Aut(S, C)-vertex transitivity

12.3 Application to strong semilattices of left groups

Application to strong semilattices of groups

12.4 End’ (S, C)-vertex transitive Cayley graphs

12.5 Comments

13 Embeddings of Cayley graphs – genus of semigroups

13.1 The genus of a group

13.2 Toroidal right groups

13.3 The genus of (A × Rr)

Cayley graphs of A × R4

Constructions of Cayley graphs for A × R2 and A × R3

13.4 Non-planar Clifford semigroups

13.5 Planar Clifford semigroups

13.6 Comments

Bibliography

Index

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Tags: Ulrich Knauer, Algebraic Graph, Morphisms, Monoids, Matrices