Looking for a good book on groups and graph theory

Question

Can anyone help me on which book would be best book to understand

Permutation groups how it represents graphs in graph theory and other with good simple examples and is much self explanatory

And dihedral groups how it represents graphs too

Orbits and stabilizer

In a very self explanatory way with examples relating it to graphs in graph theory with examples from basis

For a beginner

Kindly help by recommending a book that I can study as a beginner to learn algebra in graph theory of the above topic mainly permutation groups, orbits and stabilizer

Answer 1

I'm not entirely sure if this is what you are looking for, but there is a field known as "Geometric Group Theory" where we study groups by their actions on graphs (and other things too).

A fantastically approachable first reference (I'm biased, as it was my first reference) is Meier's "Groups, Graphs, and Trees".

As for how this pertains to your question, chapter 1 of this book discusses Cayley's Theorem (that every group is a subgroup of a symmetric group) and what is somewhat cheekily referred to as Cayley's Better Theorem (that every group is a group of symmetries of some graph). Chapter 1 also discusses the orbit-stabilizer theorem, as well as the dihedral groups (and how they act on graphs).

The entire book is wonderful, but already the first 2 chapters will discuss everything you have brought up in this question. It is filled with good exposition and fantastic pictures to help you see what's happening.

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I hope this helps ^_^