Modern Graph Theory

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

An overview 1. Graph theory 2. Number theory 3. PSL2(q) 4. The graphs Xp,q Appendix A. 4-regular graphs with large girth Index Bibliography.

Elementary Number Theory, Group Theory, and Ramanujan Graphs: An Overview

This is the first comprehensive monograph on the mathematical theory of the solitaire game The Tower of Hanoi which was invented in the 19th century by the French number theorist douard Lucas. The book comprises a survey of the historical development from the games predecessors up to recent research in mathematics and applications in computer science and psychology. Apart from long-standing myths it contains a thorough, largely self-contained presentation of the essential mathematical facts with complete proofs, including also unpublished material. The main objects of research today are the so-called Hanoi graphs and the related Sierpiski graphs. Acknowledging the great popularity of the topic in computer science, algorithms and their correctness proofs form an essential part of the book. In view of the most important practical applications of the Tower of Hanoi and its variants, namely in physics, network theory, and cognitive (neuro)psychology, other related structures and puzzles like, e.g., the Tower of London, are addressed. Numerous captivating integer sequences arise along the way, but also many open questions impose themselves. Central among these is the famed Frame-Stewart conjecture. Despite many attempts to decide it and large-scale numerical experiments supporting its truth, it remains unsettled after more than 70 years and thus demonstrates the timeliness of the topic. Enriched with elaborate illustrations, connections to other puzzles and challenges for the reader in the form of (solved) exercises as well as problems for further exploration, this book is enjoyable reading for students, educators, game enthusiasts and researchers alike.

https://link.springer.com/content/pdf/10.1007/978-3-319-73779-9.pdf

The Tower of Hanoi – Myths and Maths

Review: Walter J. Savitch, Relationships between Nondeterministic and Deterministic Tape Complexities

To any self-similar action of a finitely generated group $G$ of
automorphisms of a regular rooted tree $T$ can be naturally
associated an infinite sequence of finite graphs
$\{\Gamma_n\}_{n\geq 1}$, where $\Gamma_n$ is the Schreier graph
of the action of $G$ on the $n$-th level of $T$. Moreover, the
action of $G$ on $\partial T$ gives rise to orbital Schreier
graphs $\Gamma_{\xi}$, $\xi\in \partial T$. Denoting by $\xi_n$
the prefix of length $n$ of the infinite ray $\xi$, the rooted
graph $(\Gamma_{\xi},\xi)$ is then the limit of the sequence of
finite rooted graphs $\{(\Gamma_n,\xi_n)\}_{n\geq 1}$ in the sense
of pointed Gromov-Hausdorff convergence. In this paper, we give a
complete classification (up to isomorphism) of the limit graphs
$(\Gamma_{\xi},\xi)$ associated with the Basilica group acting on
the binary tree, in terms of the infinite binary sequence
$\xi$.

/pdf/schreier-graphs-of-the-basilica-group-m40basm6m7.pdf

Schreier graphs of the Basilica group

We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk’s group of intermediate growth; the iterated monodromy group of the complex polynomial z 2-1 known as the “Basilica group”; and the Hanoi Towers group H (3) closely related to the Sierpinski gasket.

Partition Functions of the Ising Model on Some Self-similar Schreier Graphs

We study partition functions for the dimer model on families of finite graphs converging to infinite self-similar graphs and forming approximation sequences to certain well-known fractals. The graphs that we consider are provided by actions of finitely generated groups by automorphisms on rooted trees, and thus their edges are naturally labeled by the generators of the group. It is thus natural to consider weight functions on these graphs taking different values according to the labeling. We study in detail the well-known example of the Hanoi Towers group H ( 3 ) , closely related to the Sierpinski gasket.

Counting dimer coverings on self-similar Schreier graphs

We compute the complexity of two infinite families of finite graphs: the Sierpinski graphs, which are finite approximations of the well-known Sierpinski gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we study the weighted generating functions of the spanning trees, associated with several natural labellings of the edge sets.

/pdf/weighted-spanning-trees-on-some-self-similar-graphs-17h1nskzht.pdf

Weighted Spanning Trees on some Self-Similar Graphs

In this work we define two kinds of crested product for reversible Markov chains, which naturally appear as a generalization of the case of crossed and nested product, as in association schemes theory, even if we do a construction that seems to be more general and simple. Although the crossed and nested product are inspired by the study of Gelfand pairs associated with the direct and the wreath product of two groups, the crested products are a more general construction, independent from the Gelfand pairs theory, for which a complete spectral theory is developed. Moreover, the k-step transition probability is given. It is remarkable that these Markov chains describe some classical models (Ehrenfest diffusion model, Bernoulli–Laplace diffusion model, exclusion model) and give some generalization of them. As a particular case of nested product, one gets the classical Insect Markov chain on the ultrametric space. Finally, in the context of the second crested product, we present a generalization of this Markov chain to the case of many insects and give the corresponding spectral decomposition.

https://archive-ouverte.unige.ch/unige:8517/ATTACHMENT01

Daniele D'Angeli

Papers

Schreier graphs of the Basilica group

Partition Functions of the Ising Model on Some Self-similar Schreier Graphs

Counting dimer coverings on self-similar Schreier graphs

Weighted Spanning Trees on some Self-Similar Graphs

Crested products of Markov chains