$PI$-eigenfunctions of the Star graphs

TL;DR

This paper studies the eigenfunctions of the Star graph, a Cayley graph of the symmetric group generated by transpositions, revealing new families of eigenfunctions and their relation to Specht modules, with implications for eigenfunction reconstruction.

Contribution

It introduces a family of $PI$-eigenfunctions for the Star graph with specific eigenvalues and links these functions to Specht modules, advancing understanding of the graph's spectral properties.

Findings

Constructed $PI$-eigenfunctions with eigenvalue $n-m-1$

Connected eigenfunctions to Specht module basis

Reconstructed eigenfunctions from second neighborhood for eigenvalue $n-2$

Abstract

We consider the symmetric group $\mathrm{Sym}_n,\,n\geqslant 2$, generated by the set $S$ of transpositions $(1~i),\,2 \leqslant i \leqslant n$, and the Cayley graph $S_n=Cay(\mathrm{Sym}_n,S)$ called the Star graph. For any positive integers $n\geqslant 3$ and $m$ with $n > 2m$, we present a family of $PI$-eigenfunctions of $S_n$ with eigenvalue $n-m-1$. We establish a connection of these functions with the standard basis of a Specht module. In the case of largest non-principal eigenvalue $n-2$ we prove that any eigenfunction of $S_n$ can be reconstructed by its values on the second neighbourhood of a vertex.