Anti-regular graphs with loops and their spectrum

TL;DR

This paper characterizes a special class of graphs with loops and unique degree sequences, determining their adjacency spectrum using orthogonal polynomial techniques, and extends the concept of anti-regular graphs.

Contribution

It introduces the spectral characterization of anti-regular graphs with loops, expanding the understanding of their structure and spectral properties.

Findings

Graphs with loops and unique degree sequences are characterized.

The adjacency spectrum of these graphs is explicitly determined.

The method involves orthogonal polynomials and tridiagonal matrices.

Abstract

We characterize the graphs with loops whose degree sequences have no repeated values and find their adjacency spectrum. In the case of simple graphs, such graphs are called anti-regular graphs and are examples of threshold graphs. The spectrum is found by using the known relationship between tridiagonal matrices and orthogonal polynomials.