2 research outputs found
Graphs with three and four distinct eigenvalues based on circulants
In this paper, we aim to address the open questions raised in various recent
papers regarding characterization of circulant graphs with three or four
distinct eigenvalues in their spectra. Our focus is on providing
characterizations and constructing classes of graphs falling under this
specific category. We present a characterization of circulant graphs with prime
number order and unitary Cayley graphs with arbitrary order, both of which
possess spectra displaying three or four distinct eigenvalues. Various
constructions of circulant graphs with composite orders are provided whose
spectra consist of four distinct eigenvalues. These constructions primarily
utilize specific subgraphs of circulant graphs that already possess two or
three eigenvalues in their spectra, employing graph operations like the tensor
product, the union, and the complement. Finally, we characterize the iterated
line graphs of unitary Cayley graphs whose spectra contain three or four
distinct eigenvalues, and we show their non-circulant nature.Comment: 24 page