Spectrum of some arrow-bordered circulant matrix

Abstract

Given a circulant matrix circ(c, a, 0, 0, ..., 0, a), a = 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 abc matrix , or an 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.