Spectrum of some arrow-bordered circulant matrix

Given a circulant matrix $\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}(c,a,0,0,\mathrm{.}\mathrm{.}\mathrm{.},0,a)$, $a\mathrm{\ne }0$, of order~$n$, we ``border’’ it from left and from above by constant column and row, respectively, and we set the left top entry to be $-nc$. This way we get a~particular title object, an example of what we call an $\mathit{\text{abc}}\mathit{\text{matrix}}\mathit{\text{/}}$, or an $\mathit{\text{arrow-bordered}}\mathit{\text{circulant}}\mathit{\text{(matrix)}}\mathit{\text{/}}$. We find its eigenpairs and we discuss its spectrum with stress on extreme eigenvalues and their bounds. At last we notice its relation to a~weighted wheel graph