Bounds on the Parameters of Non-L-Borderenergetic Graphs

We consider graphs whose Laplacian energy is equivalent to the Laplacian energy of the complete graph of the same order, which is called an L-borderenergetic graph. First, we study the graphs with degree sequence consisting of at most three distinct integers and give new bounds for the number of vertices of these graphs to be non-L-borderenergetic. Second, by using Koolen–Moulton and McClelland inequalities, we give new bounds for the number of edges of a non-L-borderenergetic graph. Third, we use recent bounds established by Milovanovic, et al. for the Laplacian energy to get similar conditions for non-L-borderenergetic graphs. Our bounds depend only on the degree sequence of a graph, which is much easier than computing the spectrum of the graph. In other words, we develop a faster approach to exclude non-L-borderenergetic graphs.