-borderenergetic graphs. Issue 1 (2nd January 2020) - Tao, Qingyun; Hou, Yaoping
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 K n . 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 K n, i.e., Q E ( G ) = Q E ( K n ) = 2 n − 2 . In this paper, we construct some infinite family of graphs satisfying Q E ( G ) = L E ( G ) = 2 n − 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 Q E ( G ) = L E ( 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.