On Laplacian energy in terms of graph invariants

Open Problem on $\sigma$-invariant

“…This is another motivation to study this graph invariant for the Laplacian energy of a graph G. Therefore, the spectral parameter σ is reasonable relevant in spectral graph theory. To know more information about this spectral graph invariant and its applications, see [7,8,19]. In [7], all graphs with σ(G) = n − 1 were characterized and the result was applied for Laplacian energy of graphs.…”

“…To know more information about this spectral graph invariant and its applications, see [7,8,19]. In [7], all graphs with σ(G) = n − 1 were characterized and the result was applied for Laplacian energy of graphs. It is interesting to characterize all graphs for some specific value of σ = σ(G) between 1 and n − 2.…”

A Conjecture on Laplacian Eigenvalues of Trees

“…There is evidence that σ(G) plays an important role in defining structural properties of a graph G. For example, it is related to the clique number ω of G (the number of vertices of the largest induced complete subgraph of G) and it also gives insight about the Laplacian energy of a graph [5,19]. Moreover, several structural properties of a graph are related to σ (see, for example [4,5]).…”

Partial Characterization of Graphs Having a Single Large Laplacian Eigenvalue

“…There is evidence that σ(G) plays an important role in defining structural properties of a graph G. For example, it is related to the clique number ω of G (the number of vertices of the largest induced complete subgraph of G) and it also gives insight about the Laplacian energy of a graph [15,3]. Moreover, several structural properties of a graph are related to σ (see, for example [2,3]).…”