Laplacian energy and first Zagreb index of Laplacian integral graphs

The set S_i,n = {0, 1, 2, …, i − 1, i + 1, …, n − 1, n}, 1 ⩽ i ⩽ n, is called Laplacian realizable if there exists a simple connected undirected graph whose Laplacian spectrum is S_i,n. The existence of such graphs was established by S. Fallat et all. In the present paper, we find the Laplacian energy and first Zagreb index of graphs whose Laplacian spectrum is S_i,n.