Formula omitted]-borderenergetic graphs

Abstract

A graph G is said to be borderenergetic (L-borderenergetic, respectively) if its energy (Laplacian energy, respectively) equals the energy (Laplacian energy, respectively) of the complete graph Kn. We extend this concept to signless Laplacian energy of a graph. A graph G is called Q-borderenergetic if its signless Laplacian energy is same as that of the complete graph Kn, i.e., QE(G)=QE(Kn)=2n−2. In this paper, we construct some infinite family of graphs satisfying QE(G)=LE(G)=2n−2, this happens to give a positive answer to the open problem mentioned by Nair Abreu et al. in Nair Abreu et al. (2011), that is whether there are connected non-bipartite, non-regular graphs satisfying QE(G)=LE(G). We also obtain some bounds on the order and size of Q-borderenergetic graphs. Finally, we use a computer search to find out all Q-borderenergetic graphs on no more than 10 vertices, the number of such graphs is 39.