Babai Numbers and Babai Spectra of Paths and Cycles

TL;DR

This paper provides a complete characterization of Babai numbers and Babai spectra for paths and cycles, offering precise mathematical results for these graph classes.

Contribution

It fully determines Babai numbers and spectra for paths and cycles, filling gaps in the understanding of these graph invariants.

Findings

Babai numbers of paths are explicitly determined for all n>1.

Babai k-spectra of paths are characterized for 1 ≤ k ≤ n/2.

Babai numbers and spectra of cycles are fully characterized for specified k and n.

Abstract

We study Babai numbers and Babai $k$-spectra of paths and cycles. We completely determine the Babai numbers of paths $P_n$ for $n>1$ and $1 \leq k \leq n-1$, and the Babai $k$-spectra for $P_n$ when $1 \leq k \leq n/2$. We also completely determine Babai numbers and Babai $k$-spectra of all cycles $C_n$ for $k \in \{1,2\}$ and $n \geq 3$ if $k=1$ and $n > 3$ if $k=2$.