Construction for the sequences of \(Q\)-borderenergetic graphs (Q783164)

Summary: This research intends to construct a signless Laplacian spectrum of the complement of any \(k\)-regular graph \(G\) with order \(n\). Through application of the join of two arbitrary graphs, a new class of \(Q\)-borderenergetic graphs is determined with proof. As indicated in the research, with a regular \(Q\)-borderenergetic graph, sequences of regular \(Q\)-borderenergetic graphs can be constructed. The procedures for such a construction are determined and demonstrated. Significantly, all the possible regular \(Q\)-borderenergetic graphs of order \(7<n\leq10\) are determined.