Minimum supports of eigenfunctions with the second largest eigenvalue of the Star graph

TL;DR

This paper investigates the minimum support size of eigenfunctions associated with the second largest eigenvalue of the star graph, providing exact values and characterizations for certain graph sizes.

Contribution

It determines the minimum support size and characterizes eigenfunctions for the second largest eigenvalue of star graphs for specific values of n.

Findings

Minimum support size for eigenfunctions when n ≥ 8 and n=3

Characterization of eigenfunctions with minimum support

Exact support sizes for specific star graph cases

Abstract

The Star graph $S_n$, $n\ge 3$, is the Cayley graph on the symmetric group $Sym_n$ generated by the set of transpositions $\{(12),(13),\ldots,(1n)\}$. In this work we study eigenfunctions of $S_n$ corresponding to the second largest eigenvalue $n-2$. For $n\ge 8$ and $n=3$, we find the minimum cardinality of the support of an eigenfunction of $S_n$ corresponding to the second largest eigenvalue and obtain a characterization of eigenfunctions with the minimum cardinality of the support.