Some Thoughts on Symmetries of Graphs

The algebraic graph theory course that I am taking uses the symmetries of graphs to study certain properties about graphs. But a part of me also hope that studying these graph symmetries will also give me better insight to various finite groups, mainly the symmetric groups . While we can see S₃ as the symmetries of the equilateral triangle and S₄ as the rotational group of the cube, it has been difficult for me to visualize S_n of higher value n as the symmetry group of objects with a certain kind of structure.

Having a better understanding of graphs could be a new way for me to understand these symmetry groups. And it will change the way that I see symmetry. Because to be honest, the first time I encountered graphs, I was not that huge of a fan. I found graphs to be lacking something that made looking at them not all that satisfying… Their physical form is mercurial and their symmetries are not the tangible and apparent symmetries that I encounter in polygons and platonic solids.

However, after this post: https://menagerieofmathematics.wordpress.com/2025/01/15/thoughts-on-re-reading-chapter-1-of-arnolds-ode/

I’m starting to understand symmetries differently now. Symmetries are a part of our perception and experience. Just because I have been accustomed to seeing symmetries through polygons and platonic solids does not mean all symmetries are like; I just need to learn to find and see them because they are no longer so apparent.