scholarly journals On Laplacian Equienergetic Signed Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Qingyun Tao ◽  
Lixin Tao
Keyword(s):  
Average Degree ◽  
Signed Graph ◽  
Signed Graphs ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy

The Laplacian energy of a signed graph is defined as the sum of the distance of its Laplacian eigenvalues from its average degree. Two signed graphs of the same order are said to be Laplacian equienergetic if their Laplacian energies are equal. In this paper, we present several infinite families of Laplacian equienergetic signed graphs.

scholarly journals Inequalities for Laplacian Eigenvalues of Signed Graphs with Given Frustration Number

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1902
Author(s):  
Milica Anđelić ◽  
Tamara Koledin ◽  
Zoran Stanić
Keyword(s):  
Social Groups ◽  
Laplacian Matrix ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimal Set ◽  
Laplacian Eigenvalue ◽  
Laplacian Eigenvalues ◽  
Negative Edge ◽  
Average Vertex ◽  
Symmetric Relations

Balanced signed graphs appear in the context of social groups with symmetric relations between individuals where a positive edge represents friendship and a negative edge represents enmities between the individuals. The frustration number f of a signed graph is the size of the minimal set F of vertices whose removal results in a balanced signed graph; hence, a connected signed graph G˙ is balanced if and only if f=0. In this paper, we consider the balance of G˙ via the relationships between the frustration number and eigenvalues of the symmetric Laplacian matrix associated with G˙. It is known that a signed graph is balanced if and only if its least Laplacian eigenvalue μn is zero. We consider the inequalities that involve certain Laplacian eigenvalues, the frustration number f and some related invariants such as the cut size of F and its average vertex degree. In particular, we consider the interplay between μn and f.

scholarly journals On a conjecture of Laplacian energy of trees

2021 ◽  
pp. 2250009
Author(s):  
Hilal A. Ganie ◽  
Bilal A. Rather ◽  
S. Pirzada
Keyword(s):  
Sufficient Conditions ◽  
Simple Graph ◽  
Average Degree ◽  
Laplacian Eigenvalues ◽  
Path Graph ◽  
Laplacian Energy ◽  
The Family ◽  
Pendent Vertices

Let [Formula: see text] be a simple graph with [Formula: see text] vertices, [Formula: see text] edges having Laplacian eigenvalues [Formula: see text]. The Laplacian energy LE[Formula: see text] is defined as LE[Formula: see text], where [Formula: see text] is the average degree of [Formula: see text]. Radenković and Gutman conjectured that among all trees of order [Formula: see text], the path graph [Formula: see text] has the smallest Laplacian energy. Let [Formula: see text] be the family of trees of order [Formula: see text] having diameter [Formula: see text]. In this paper, we show that Laplacian energy of any tree [Formula: see text] is greater than the Laplacian energy of [Formula: see text], thereby proving the conjecture for all trees of diameter [Formula: see text]. We also show the truth of conjecture for all trees with number of non-pendent vertices at most [Formula: see text]. Further, we give some sufficient conditions for the conjecture to hold for a tree of order [Formula: see text].

scholarly journals The Laplacian Energy of Conjugacy Class Graph of Some Finite Groups

MATEMATIKA ◽  
2019 ◽  
Vol 35 (1) ◽  
pp. 59-65
Author(s):  
Rabiha Mahmoud ◽  
Amira Fadina Ahmad Fadzil ◽  
Nor Haniza Sarmin ◽  
Ahmad Erfanian
Keyword(s):  
Conjugacy Class ◽  
Finite Groups ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Laplacian Eigenvalues ◽  
Laplacian Matrices ◽  
Laplacian Energy ◽  
The Difference ◽  
Generalized Quaternion ◽  
Class Graph

Let G be a dihedral group and its conjugacy class graph. The Laplacian energy of the graph, is defined as the sum of the absolute values of the difference between the Laplacian eigenvalues and the ratio of twice the edges number divided by the vertices number. In this research, the Laplacian matrices of the conjugacy class graph of some dihedral groups, generalized quaternion groups, quasidihedral groups and their eigenvalues are first computed. Then, the Laplacian energy of the graphs are determined.

scholarly journals On relations between Kirchhoff index, Laplacian energy, Laplacian-energy-like invariant and degree deviation of graphs

Filomat ◽  
2020 ◽  
Vol 34 (3) ◽  
pp. 1025-1033
Author(s):  
Predrag Milosevic ◽  
Emina Milovanovic ◽  
Marjan Matejic ◽  
Igor Milovanovic
Keyword(s):  
Topological Index ◽  
Degree Sequence ◽  
Laplacian Matrix ◽  
Vertex Degree ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Degree Deviation ◽  
Simple Connected Graph

Let G be a simple connected graph of order n and size m, vertex degree sequence d1 ? d2 ?...? dn > 0, and let ?1 ? ? 2 ? ... ? ?n-1 > ?n = 0 be the eigenvalues of its Laplacian matrix. Laplacian energy LE, Laplacian-energy-like invariant LEL and Kirchhoff index Kf, are graph invariants defined in terms of Laplacian eigenvalues. These are, respectively, defined as LE(G) = ?n,i=1 |?i-2m/n|, LEL(G) = ?n-1 i=1 ??i and Kf (G) = n ?n-1,i=1 1/?i. A vertex-degree-based topological index referred to as degree deviation is defined as S(G) = ?n,i=1 |di- 2m/n|. Relations between Kf and LE, Kf and LEL, as well as Kf and S are obtained.

On skew Laplacian spectrum and energy of digraphs

2020 ◽  
pp. 2150051 ◽  
Author(s):  
Hilal A. Ganie ◽  
S. Pirzada ◽  
Bilal A. Chat ◽  
X. Li
Keyword(s):  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Arbitrary Direction ◽  
Laplacian Eigenvalues ◽  
Underlying Graph ◽  
Laplacian Energy ◽  
Energy Formula

We consider the skew Laplacian matrix of a digraph [Formula: see text] obtained by giving an arbitrary direction to the edges of a graph [Formula: see text] having [Formula: see text] vertices and [Formula: see text] edges. With [Formula: see text] to be the skew Laplacian eigenvalues of [Formula: see text], the skew Laplacian energy [Formula: see text] of [Formula: see text] is defined as [Formula: see text]. In this paper, we analyze the effect of changing the orientation of an induced subdigraph on the skew Laplacian spectrum. We obtain bounds for the skew Laplacian energy [Formula: see text] in terms of various parameters associated with the digraph [Formula: see text] and the underlying graph [Formula: see text] and we characterize the extremal digraphs attaining these bounds. We also show these bounds improve some known bounds for some families of digraphs. Further, we show the existence of some families of skew Laplacian equienergetic digraphs.

On distance Laplacian spectrum (energy) of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050061 ◽  
Author(s):  
Hilal A. Ganie
Keyword(s):  
Chromatic Number ◽  
Wiener Index ◽  
Laplacian Spectrum ◽  
Connected Graphs ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Number Formula ◽  
Energy Formula ◽  
Average Transmission ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] of order [Formula: see text] having distance Laplacian eigenvalues [Formula: see text], the distance Laplacian energy [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the Wiener index of [Formula: see text]. We obtain the distance Laplacian spectrum of the joined union of graphs [Formula: see text] in terms of their distance Laplacian spectrum and the spectrum of an auxiliary matrix. As application, we obtain the distance Laplacian spectrum of the lexicographic product of graphs. We study the distance Laplacian energy of connected graphs with given chromatic number [Formula: see text]. We show that among all connected graphs with chromatic number [Formula: see text] the complete [Formula: see text]-partite graph has the minimum distance Laplacian energy. Further, we discuss the distribution of distance Laplacian eigenvalues around average transmission degree [Formula: see text].

scholarly journals Characterizing attitudinal network graphs through frustration cloud

2021 ◽  
Author(s):  
Lucas Rusnak ◽  
Jelena Tešić
Keyword(s):  
Social Network ◽  
Survey Data ◽  
Spectral Clustering ◽  
Signed Graph ◽  
Global Measure ◽  
Signed Graphs ◽  
Community Discovery ◽  
Scalable Algorithm ◽  
Signed Network ◽  
Zero Sum

AbstractAttitudinal network graphs are signed graphs where edges capture an expressed opinion; two vertices connected by an edge can be agreeable (positive) or antagonistic (negative). A signed graph is called balanced if each of its cycles includes an even number of negative edges. Balance is often characterized by the frustration index or by finding a single convergent balanced state of network consensus. In this paper, we propose to expand the measures of consensus from a single balanced state associated with the frustration index to the set of nearest balanced states. We introduce the frustration cloud as a set of all nearest balanced states and use a graph-balancing algorithm to find all nearest balanced states in a deterministic way. Computational concerns are addressed by measuring consensus probabilistically, and we introduce new vertex and edge metrics to quantify status, agreement, and influence. We also introduce a new global measure of controversy for a given signed graph and show that vertex status is a zero-sum game in the signed network. We propose an efficient scalable algorithm for calculating frustration cloud-based measures in social network and survey data of up to 80,000 vertices and half-a-million edges. We also demonstrate the power of the proposed approach to provide discriminant features for community discovery when compared to spectral clustering and to automatically identify dominant vertices and anomalous decisions in the network.

scholarly journals Energy and Laplacian energy of unitary addition Cayley graphs

Filomat ◽  
2019 ◽  
Vol 33 (11) ◽  
pp. 3599-3613
Author(s):  
Naveen Palanivel ◽  
A.V. Chithra
Keyword(s):  
Prime Number ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Prime Factors

In this paper, we obtain the eigenvalues and Laplacian eigenvalues of the unitary addition Cayley graph Gn and its complement. Moreover, we compute the bounds for energy and Laplacian energy for Gn and its complement. In addition, we prove that Gn is hyperenergetic if and only if n is odd other than the prime number and power of 3 or n is even and has at least three distinct prime factors. It is also shown that the complement of Gn is hyperenergetic if and only if n has at least two distinct prime factors and n ? 2p.

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