Almost borderenergetic line graphs

The energy of a graph is calculated by summing the absolute values of the eigenvalues found in its adjacency matrix. In this study, we present examples of line graphs with energy equivalent to the energy of a complete graph, which are called the borderenergetic graphs. As the examples of borderenergetic line graphs are rare, we introduce a new type of graphs called almost borderenergetic graphs whose energy differs from the borderenergetic energy by at most 1. In this paper, we begin by exploring the energy properties of line graphs derived from regular graphs and strongly regular graphs. Specifically, we establish a criterion for a line graph to exhibit borderenergetic characteristics when it consists of p many connected regular graphs and q many complete graphs, even when the original regular graph is disconnected. Additionally, we present a criterion for the line graph of a strongly regular graph to exhibit borderenergetic characteristics. To illustrate these concepts, we offer examples of connected borderenergetic graphs that are not necessarily complete. In the second part, we give the spectrum of the line graph of rK1∇K2, and show that it is an almost borderenergetic graph for r ≥ 5.