scholarly journals Further Results on the Nullity of Signed Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yu Liu ◽  
Lihua You
Keyword(s):  
Characteristic Polynomial ◽  
Signed Graph ◽  
Signed Graphs ◽  
Cut Points ◽  
Eigenvalue Zero

The nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum. A signed graph is a graph with a sign attached to each of its edges. In this paper, we apply the coefficient theorem on the characteristic polynomial of a signed graph and give two formulae on the nullity of signed graphs with cut-points. As applications of the above results, we investigate the nullity of the bicyclic signed graphΓ∞p,q,l, obtain the nullity set of unbalanced bicyclic signed graphs, and thus determine the nullity set of bicyclic signed graphs.

Polynomial reconstruction of signed graphs whose least eigenvalue is close to -2

2016 ◽  
Vol 31 ◽  
pp. 740-753 ◽  
Author(s):  
Slobodan Simić ◽  
Zoran Stanic
Keyword(s):  
Characteristic Polynomial ◽  
Reconstruction Problem ◽  
Signed Graphs ◽  
Simple Graphs ◽  
Least Eigenvalue ◽  
Polynomial Reconstruction ◽  
Special Classes

The polynomial reconstruction problem for simple graphs has been considered in the literature for more than forty years and is not yet resolved except for some special classes of graphs. Recently, the same problem has been put forward for signed graphs. Here, the reconstruction of the characteristic polynomial of signed graphs whose vertex-deleted subgraphs have least eigenvalue greater than $-2$ is considered.

scholarly journals Signed graphs with cut points whose positive inertia indexes are two

2018 ◽  
Vol 539 ◽  
pp. 14-27 ◽  
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Fenglei Tian
Keyword(s):  
Signed Graphs ◽  
Cut Points

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 Restructured class of estimators for population mean using an auxiliary variable under simple random sampling scheme

2021 ◽  
Vol 17 (2) ◽  
pp. 75-90
Author(s):  
B. Prashanth ◽  
K. Nagendra Naik ◽  
R. Salestina M
Keyword(s):  
Characteristic Polynomial ◽  
Random Sampling ◽  
Simple Random Sampling ◽  
Auxiliary Variable ◽  
Sampling Scheme ◽  
Signed Graph ◽  
Population Mean ◽  
Class Of Estimators

Abstract With this article in mind, we have found some results using eigenvalues of graph with sign. It is intriguing to note that these results help us to find the determinant of Normalized Laplacian matrix of signed graph and their coe cients of characteristic polynomial using the number of vertices. Also we found bounds for the lowest value of eigenvalue.

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.

Independent domination in signed graphs

2020 ◽  
pp. 2150065
Author(s):  
P. Jeyalakshmi ◽  
K. Karuppasamy ◽  
S. Arockiaraj
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Independent Domination ◽  
Independent Dominating Set

Let [Formula: see text] be a signed graph. A dominating set [Formula: see text] is said to be an independent dominating set of [Formula: see text] if [Formula: see text] is a fully negative. In this paper, we initiate a study of this parameter. We also establish the bounds and characterization on the independent domination number of a signed graph.

Domination in signed graphs

2020 ◽  
pp. 2050094
Author(s):  
P. Jeyalakshmi
Keyword(s):  
Social Science ◽  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimum Cardinality ◽  
Underlying Graph

Let [Formula: see text] be a graph. A signed graph is an ordered pair [Formula: see text] where [Formula: see text] is a graph called the underlying graph of [Formula: see text] and [Formula: see text] is a function called a signature or signing function. Motivated by the innovative paper of B. D. Acharya on domination in signed graphs, we consider another way of defining the concept of domination in signed graphs which looks more natural and has applications in social science. A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if [Formula: see text] for all [Formula: see text]. The domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a dominating set of [Formula: see text]. Also, a dominating set [Formula: see text] of [Formula: see text] with [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we initiate a study on this parameter.

Critical concepts of restrained domination in signed graphs

2021 ◽  
pp. 2250010
Author(s):  
Anisha Jean Mathias ◽  
V. Sangeetha ◽  
Mukti Acharya
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimum Cardinality ◽  
Underlying Graph ◽  
Restrained Domination ◽  
Edge Removal ◽  
Restrained Domination Number

A signed graph [Formula: see text] is a simple undirected graph in which each edge is either positive or negative. Restrained dominating set [Formula: see text] in [Formula: see text] is a restrained dominating set of the underlying graph [Formula: see text] where the subgraph induced by the edges across [Formula: see text] and within [Formula: see text] is balanced. The minimum cardinality of a restrained dominating set of [Formula: see text] is called the restrained domination number, denoted by [Formula: see text]. In this paper, we initiate the study on various critical concepts to investigate the effect of edge removal or edge addition on restrained domination number in signed graphs.

On The Unitary Cayley Ring Signed Graphs$S_n^\oplus$

2013 ◽  
Vol 14 (04) ◽  
pp. 1350020 ◽  
Author(s):  
DEEPA SINHA ◽  
AYUSHI DHAMA
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Vertex Set ◽  
Unitary Cayley Graph

A Signed graph (or sigraph in short) is an ordered pair S = (G, σ), where G is a graph G = (V, E) and σ : E → {+, −} is a function from the edge set E of G into the set {+, −}. For a positive integer n > 1, the unitary Cayley graph Xnis the graph whose vertex set is Zn, the integers modulo n and if Undenotes set of all units of the ring Zn, then two vertices a, b are adjacent if and only if a − b ∈ Un. In this paper, we have obtained a characterization of balanced and clusterable unitary Cayley ring sigraph [Formula: see text]. Further, we have established a characterization of canonically consistent unitary Cayley ring sigraph [Formula: see text], where n has at most two distinct odd primes factors. Also sign-compatibility has been worked out for the same.

Switched signed graphs of integer additive set-valued signed graphs

2017 ◽  
Vol 09 (04) ◽  
pp. 1750043 ◽  
Author(s):  
N. K. Sudev ◽  
K. P. Chithra ◽  
K. A. Germina
Keyword(s):  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
Power Set ◽  
Injective Map ◽  
Signature Formula

Let [Formula: see text] denote a set of non-negative integers and [Formula: see text] be its power set. An integer additive set-labeling (IASL) of a graph [Formula: see text] is an injective set-valued function [Formula: see text] such that the induced function [Formula: see text] is defined by [Formula: see text], where [Formula: see text] is the sumset of [Formula: see text] and [Formula: see text]. An IASL of a signed graph [Formula: see text] is an IASL of its underlying graph [Formula: see text] together with the signature [Formula: see text] defined by [Formula: see text]. A marking of a signed graph is an injective map [Formula: see text], defined by [Formula: see text] for all [Formula: see text]. Switching of signed graph is the process of changing the sign of the edges in [Formula: see text] whose end vertices have different signs. In this paper, we discuss certain characteristics of the switched signed graphs of certain types of integer additive set-labeled signed graphs.

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