# Some Algebraic Properties of Sierpi\'nski-Type Graphs

## TL;DR

This paper explores algebraic properties of Sierpiński graphs and their generalizations, including spectra, Cayley graph characterization, and introduces new non-Cayley vertex-transitive graphs and conjectures on Laplacian spectra.

## Contribution

It provides the spectrum and Cayley graph characterization for regular generalized Sierpiński graphs, and introduces new non-Cayley vertex-transitive graphs.

## Key findings

- Spectrum of regular generalized Sierpiński graphs determined
- Characterization of Cayley graphs within this family
- New non-Cayley vertex-transitive graphs and non-Cayley numbers

## Abstract

This paper deals with some of the algebraic properties of Sierpi\'nski graphs and a family of regular generalized Sierpi\'nski graphs. For the family of regular generalized Sierpi\'nski graphs, we obtain their spectrum and characterize those graphs that are Cayley graphs. As a by-product, a new family of non-Cayley vertex-transitive graphs, and consequently, a new set of non-Cayley numbers are introduced. We also obtain the Laplacian spectrum of Sierpi\'nski graphs in some particular cases, and make a conjecture on the general case.