On the Existence of Non-Complete L-Borderenergetic Graphs

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. A graph G on n vertices is said to be borderenergetic if its energy equals to the energy of the complete graph K-n. In [12], Tura promote this concept for Laplacian matrices. The Laplacian energy of G, introduced by Gutman and Zhou [5], is given by LE(G) = Sigma(n)(i=1) vertical bar mu(i) - (d) over bar vertical bar, where it, are the Laplacian eigenvalues of G. and (7, is the average degree of G. A graph G on n vertices is said to be L-borderenergetic if LE(G) = LE(K-n). In this paper, we first present all nqn-complete L-borderenergetic graphs of order 4, 5, 6, 7. Then we construct one connected non-complete L-borderenergetic graph on n vertices for each integer n >= 4, which extends the result in [12] and completely confirms the existence of non-complete L-borderenergetic graphs. Particularly, we prove that there are at least n/2 + 4 non-complete L-borderenergetic graphs of order n for any even integer n >= 6.