Babai Numbers and Babai Spectra of Paths and Cycles

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