Open Problem on $\sigma$-invariant

Abstract

Let G be a graph of order n with m edges. Also letbe the largest positive integer such that µ σ ≥ 2m/n. In this paper, we prove that µ 2 (G) ≥ 2m/n for almost all graphs. Moreover, we characterize the extremal graphs for any graphs. Finally, we provide the answer to Problem 3 in [8], that is, the characterization of all graphs with σ = 1.

On graphs with minimal distance signless Laplacian energy

“…The graph invariant σ , that is, the number of distance signless Laplacian eigenvalues which are greater or equal to 2W(G) n is an interesting graph invariant. Several papers exist in the literature in this direction and various open problems were asked in case of Laplacian [7] and signless Laplaian matrices. The same is true for distance signless Laplacian matrix and attractive problems of σ can be investigated, like characterization of graphs having σ = 1, 2, n 2 and σ = n − 1.…”

On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

“…where σ is the number of Laplacian eigenvalues greater than or equal to the average degree d. We note that k ∑ 1=i µ i is actually the Ky Fan k-norm, which for positive semi-definite matrices is the sum of k largest eigenvalues. The parameter σ is an active component of the present research and some work mostly on trees can be found in the literature [24]. In fact, it is shown in [25] that the Laplacian energy has remarkable chemical applications beyond the molecular orbital theory of conjugated molecules.…”