On the spectrum, energy and Laplacian energy of graphs with self-loops

The total energy of a conjugated molecule's π-electrons is a quantum-theoretical feature that has been known since the 1930s. It is determined using the Huckel tight-binding molecular orbital (HMO) method. In 1978, a modified definition of the total π-electron energy was introduced, which is now known as the graph energy. It is calculated by summing the absolute values of the eigenvalues of the adjacency matrix. Quiet Recently in the year 2022, Gutman extended the concept of conjugated systems to hetero-conjugated systems which is the extension of ordinary graph energy to energy of graph with self loops. Let $$ be an order (vertices) 'p' graph with 'q' edges and σ- self loops. The adjacency matrix of $$ is defined by $$ if $$ then $$; if $$ where $$ then $$ and zero otherwise, where $$ represents the set of all vertices with loops. Then the energy of graph with self loops is defined as $$. In this paper, we aim to analyze the adjacency and Laplacian spectra of certain non-simple standard graphs that contain self-loops. We also calculate the energy and Laplacian energy of these graphs with loops. Furthermore, we derive lower bounds for the energy of any graph containing loops and develop a MATLAB algorithm to calculate these quantities for selected non-simple standard graphs with self-loops. Our study evaluates the strength of a graph by considering the presence of loops, which are edges that connect a vertex to itself. This approach accounts for the impact of each vertex on the entire structure of the graph. By analyzing the energy of a graph with loops, we can gain a better understanding of its distinctive characteristics and behavior.