On Albertson spectral properties of graphs with self-loops

The Albertson irregularity measure is defined as $Alb(\Gamma)=\sum_{uv\in E(\Gamma)} \vert d(u)-d(v)\vert.$ In this work, the concept of Albertson energy is extended from simple graphs to graphs with self-loops. Also the expression for the Albertson eigenvalues of a graph with self-loops are given. Some bounds on the Albertson energy of graphs with self-loops and the spread of $Alb(\Gamma_S)$ are obtained. In the last section, the Albertson energy of complete, complete bipartite, crown and thorn graphs with self-loops are computed.