On sum of powers of the Laplacian eigenvalues of graphs

On the sum of powers of Laplacian eigenvalues of bipartite graphs

“…The bounds obtained here improve the results in [16] and have much simpler forms than the results in [14]. As in [16], the sum of squares of degrees appears in our bounds. Theorem 1.…”

“…Some bounds for the sum of powers of the Laplacian eigenvalues of bipartite graphs have been given in [16], [14]. The bounds obtained here improve the results in [16] and have much simpler forms than the results in [14].…”

“…Let G be a connected bipartite graph with n 3 vertices and m edges. By the proof above, we have: If α < 0 or α > 1, the lower bound in (2) is better than the one in [16]:…”

“…Let s α (G) be the sum of the αth powers of the non-zero Laplacian eigenvalues of G, where h is the number of non-zero Laplacian eigenvalues of G. The cases α = 0, 1 are trivial as s 0 (G) = h and s 1 (G) = 2m, where m is the number of edges. Various lower and upper bounds for the sum of powers of Laplacian eigenvalues have been established in [16], [17]. For a graph G, denote by Z(G) the sum of the squares of degrees of G. It is known as the first Zagreb index in mathematical chemistry [5], [15].…”

Lower bounds of the Kirchhoff and degree Kirchhoff indices

“…The bounds can be determined in terms of usual structural parameters, such as number of vertices, number of edges, vertex degrees, and similar, or extremal Laplacian and normalized Laplacian eigenvalues, or Zagreb or Randić index, etc. (see [2,3,9,10,15,16,17,20,22,23]. In this paper we consider lower bounds for some graph invariants that, in a special case, reduce to K f (G) and DK f (G).…”