Spectrum of some arrow-bordered circulant matrix

TL;DR

This paper investigates the spectrum of arrow-bordered circulant matrices, deriving their eigenvalues and eigenvectors, analyzing extreme eigenvalues, and exploring their connection to weighted wheel graphs.

Contribution

It introduces and analyzes a new class of matrices called arrow-bordered circulant matrices, providing explicit eigenpair formulas and spectral bounds.

Findings

Eigenpairs of arrow-bordered circulant matrices are explicitly derived.

Extreme eigenvalues are characterized and bounded.

The matrices are related to weighted wheel graphs.

Abstract

Given a circulant matrix $\mathrm{circ}(c,a,0,0,...,0,a)$, $a\ne 0$, 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 \textit{abc matrix\/}, or an \textit{arrow-bordered circulant (matrix)\/}. 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