The bounds of the energy and Laplacian energy of chain graphs

Mei, Yinzhen; Guo, Chengxiao; Liu, Mengtian

doi:10.3934/math.2021284

Cited by 3 publications

(4 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From (20), we obtain f (x) → +∞ as x → ∞ and f (1) = −4n + 8 + 4s 2 t 1 (s 1 + t 2 ) = 4 − (n − 2) + s 2 t 1 (n − s 2 − t 1 ) ≥ 4 (n − 4) > 0 as s 1 + s 2 + t 1 + t 2 = n and by (24). These results with (21), we obtain λ 1 ≥ λ 2 > 1. This proves Claim 1.…”

Section: Moreover Equalities Hold If and Only Ifsupporting

confidence: 73%

“…Therefore λ i < 0 for 1 ≤ i ≤ 3, or λ 1 ≥ λ 2 > 0 and λ 3 < 0. By (21), λ i < 0 (1 ≤ i ≤ 3) is not possible. Thus we have λ 1 ≥ λ 2 > 0 and λ 3 < 0.…”

Section: Moreover Equalities Hold If and Only Ifmentioning

confidence: 99%

“…Anđelić et al [4,5,13] published several papers related to spectra of chain graphs with key focus on eigenvalue location, characteristic polynomial and energy. Very recently, Mei et al [21] give some bounds of the energy and Laplacian energy of chain graphs.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the spectrum and energy of Seidel matrix for chain graphs

Мандал¹,

Mehatari²,

Das³

2022

Preprint

View full text Add to dashboard Cite

We study various spectral properties of the Seidel matrix S of a connected chain graph. We prove that −1 is always an eigenvalue of S and all other eigenvalues of S can have multiplicity at most two. We obtain the multiplicity of the Seidel eigenvalue −1, minimum number of distinct eigenvalues, eigenvalue bounds, characteristic polynomial, lower and upper bounds of Seidel energy of a chain graph. It is also shown that the energy bounds obtained here work better than the bounds conjectured by Haemers. We also obtain the minimal Seidel energy for some special chain graphs of order n. We also give a number of open problems.

show abstract

Section: Moreover Equalities Hold If and Only Ifsupporting

confidence: 73%

“…Therefore λ i < 0 for 1 ≤ i ≤ 3, or λ 1 ≥ λ 2 > 0 and λ 3 < 0. By (21), λ i < 0 (1 ≤ i ≤ 3) is not possible. Thus we have λ 1 ≥ λ 2 > 0 and λ 3 < 0.…”

Section: Moreover Equalities Hold If and Only Ifmentioning

confidence: 99%

See 1 more Smart Citation

On the spectrum and energy of Seidel matrix for chain graphs

Мандал¹,

Mehatari²,

Das³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Furthermore, Kinkar et al [7] present some lower and upper bounds on the energy of a chain graph G, as well as demonstrating that the star provides the minimal energy of connected chain graphs of order . And recently, Yinzhen et al [8] obtain some upper and lower bounds on the energy and Laplacian energy of chain graphs by comparing algebraic connectivity, where they achieve the maximal Laplacian energy among all connected bicyclic chain graphs.…”

Section: Introductionmentioning

confidence: 99%

The Energy of Conjugate Graph of a Dihedral Group

Mahmud

Tal

Idris

et al. 2021

JAMCS

View full text Add to dashboard Cite

Let \(\Gamma_{D_{2 n}}^{C}\) and \(E(\Gamma)\) denote the conjugate graph of a dihedral group of order \(2 n(n \in \aleph)\) and the energy of a graph respectively. The sum of the absolute values of the eigenvalues of an adjacency matrix's eigenvalues is the energy of a graph. In this paper, we use group representation of a dihedral group of order 2n with its conjugacy classes to explicitly design admissible conjugate graphs. We further introduced the general formula for the energy of conjugate graphs of dihedral groups in various circumstances. Also, we deduced the general formula for the conjugate graph of generalized dihedral groups of order 2n depending on the nature of n.

show abstract

New families of Laplacian borderenergetic graphs

Dede

2024

Acta Informatica

View full text Add to dashboard Cite

In the first part of this paper, we present the Laplacian spectrum of some graphs composed of complete graphs. To begin with, we give the Laplacian spectrum of G1 = aKi ∪ b(Kj × Kk) and then we find the values of i, b, j, k ∈ Z + for which G1 is Laplacian borderenergetic. Similarly, we give the Laplacian spectrum of G2 = aKi∇b(Kj × Kk) ∪ cKl and find the values of a, i, b, j, k, c, l ∈ Z + for which G2 is Laplacian borderenergetic. In this part, we construct three infinite families of non-complete disconnected L-borderenergetic graphs:Then, in the second part of this work, we construct new infinite families of noncomplete connected L-borderenergetic graphs Ω1 = {K2∇aK r 2 |a ∈ Z + }, Ω2 = {aK3 ∪ 2(K2 × K3)|a ∈ Z + } and Ω3 = {aK5 ∪ (K3 × K3)|a ∈ Z + }.

show abstract

The bounds of the energy and Laplacian energy of chain graphs

Cited by 3 publications

References 23 publications

On the spectrum and energy of Seidel matrix for chain graphs

On the spectrum and energy of Seidel matrix for chain graphs

The Energy of Conjugate Graph of a Dihedral Group

New families of Laplacian borderenergetic graphs

Contact Info

Product

Resources

About