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) = 2n − 2. In this paper, we construct some infinite family of graphs satisfying Q E(G) = L E(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 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.
The Laplacian energy of a signed graph is defined as the sum of the distance of its Laplacian eigenvalues from its average degree. Two signed graphs of the same order are said to be Laplacian equienergetic if their Laplacian energies are equal. In this paper, we present several infinite families of Laplacian equienergetic signed graphs.
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