On signed graphs with two distinct eigenvalues

Filomat ◽  
2017 ◽  
Vol 31 (20) ◽  
pp. 6393-6400 ◽  
Author(s):  
E. Ghasemian ◽  
G.H. Fath-Tabar
Keyword(s):  
Signed Graph ◽  
Signed Graphs ◽  
Sign Function ◽  
Underlying Graph ◽  
Adjacency Matrices

Let G? be a signed graph with the underlying graph G and with sign function ? : E(G) ? {?}. In this paper, we characterize the signed graphs with two distinct eigenvalues whose underlying graphs are triangle-free. Also, we classify all 3-regular and 4-regular signed graphs whose underlying graphs are triangle-free and give their adjacency matrices as well.

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.

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.

scholarly journals Restrained domination in signed graphs

2020 ◽  
Vol 12 (1) ◽  
pp. 155-163
Author(s):  
Anisha Jean Mathias ◽  
V. Sangeetha ◽  
Mukti Acharya
Keyword(s):  
Social Networks ◽  
Dominating Set ◽  
Domination Number ◽  
Important Application ◽  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
Restrained Domination ◽  
Restrained Domination Number

AbstractA signed graph Σ is a graph with positive or negative signs attatched to each of its edges. A signed graph Σ is balanced if each of its cycles has an even number of negative edges. Restrained dominating set D in Σ is a restrained dominating set of its underlying graph where the subgraph induced by the edges across Σ[D : V \ D] and within V \ D is balanced. The set D having least cardinality is called minimum restrained dominating set and its cardinality is the restrained domination number of Σ denoted by γr(Σ). The ability to communicate rapidly within the network is an important application of domination in social networks. The main aim of this paper is to initiate a study on restrained domination in the realm of different classes of signed graphs.

scholarly journals Sign-Compatibility of Some Derived Signed Graphs

2012 ◽  
Vol 11 (4) ◽  
pp. 1-14
Author(s):  
Deepa Sinha ◽  
Ayushi Dhama
Keyword(s):  
Signed Graph ◽  
Signed Graphs ◽  
Positive Edge ◽  
Underlying Graph ◽  
Negative Edge ◽  
Total Point

A signed graph (or sigraph in short) is an ordered pair S = (Su, σ), where Su is a graph G = (V, E), called the underlying graph of S and σ : E → {+1, −1} is a function from the edge set E of Su into the set {+1, −1}, called the signature of S. A sigraph S is sign-compatible if there exists a marking µ of its vertices such that the end vertices of every negative edge receive ‘−1’ marks in µ and no positive edge does so. In this paper, we characterize S such that its ×-line sigraphs, semi-total line sigraphs, semi-total point sigraphs and total sigraphs are sign-compatible.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

On Square and 2-path Signed Graph

2016 ◽  
Vol 16 (01) ◽  
pp. 1550011 ◽  
Author(s):  
DEEPA SINHA ◽  
DEEPAKSHI SHARMA
Keyword(s):  
Signed Graph ◽  
Signed Graphs ◽  
Common Edge ◽  
Underlying Graph ◽  
Square Signed Graph

A signed graph is an ordered pair [Formula: see text], where [Formula: see text] is a graph G = (V, E), called the underlying graph of S and [Formula: see text] is a function from the edge set E of Su into the set {+, -}, called the signature of S. In this paper, we characterize all those signed graphs whose 2-path signed graphs are isomorphic to their square signed graph along with algorithm to check the same. In other sections we find the characterization of signed graph S such that [Formula: see text] where D is a derived signed graph of the signed graph S such as: line signed graphs, total signed graphs, common edge signed graphs, splitting signed graphs. Also each characterization is supported by algorithms for the same.

scholarly journals Знаковые графы эквивалентности степени тоша

2020 ◽  
Author(s):  
R. Rajendra ◽  
P. S. K. Redy
Keyword(s):  
Structural Characterization ◽  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
Marked Graph ◽  
Multiple Edges ◽  
Sigma E

The Tosha-degree of an edge $\alpha $ in a graph $\Gamma$ without multiple edges, denoted by $T(\alpha)$, is the number of edges adjacent to $\alpha$ in $\Gamma$, with self-loops counted twice. A signed graph (marked graph) is an ordered pair $\Sigma=(\Gamma,\sigma)$ ($\Sigma =(\Gamma, \mu)$), where $\Gamma=(V,E)$ is a graph called the underlying graph of $\Sigma$ and $\sigma : E \rightarrow \{+,-\}$ ($\mu : V \rightarrow \{+,-\}$) is a function. In this paper, we define the Tosha-degree equivalence signed graph of a given signed graph and offer a switching equivalence characterization of signed graphs that are switching equivalent to Tosha-degree equivalence signed graphs and $ k^{th}$ iterated Tosha-degree equivalence signed graphs. It is shown that for any signed graph $\Sigma$, its Tosha-degree equivalence signed graph $T(\Sigma)$ is balanced and we offer a structural characterization of Tosha-degree equivalence signed graphs

scholarly journals The least Laplacian eigenvalue of the unbalanced unicyclic signed graphs with $k$ pendant vertices

2020 ◽  
Vol 36 (36) ◽  
pp. 390-399
Author(s):  
Qiao Guo ◽  
Yaoping Hou ◽  
Deqiong Li
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
Signed Graph ◽  
Signed Graphs ◽  
Laplacian Eigenvalue ◽  
Unicyclic Graphs ◽  
Underlying Graph ◽  
Vertex Degrees

Let $\Gamma=(G,\sigma)$ be a signed graph and $L(\Gamma)=D(G)-A(\Gamma)$ be the Laplacian matrix of $\Gamma$, where $D(G)$ is the diagonal matrix of vertex degrees of the underlying graph $G$ and $A(\Gamma)$ is the adjacency matrix of $\Gamma$. It is well-known that the least Laplacian eigenvalue $\lambda_n$ is positive if and only if $\Gamma$ is unbalanced. In this paper, the unique signed graph (up to switching equivalence) which minimizes the least Laplacian eigenvalue among unbalanced connected signed unicyclic graphs with $n$ vertices and $k$ pendant vertices is characterized.

scholarly journals Homomorphisms of planar signed graphs to signed projective cubes

2013 ◽  
Vol Vol. 15 no. 3 ◽  
Author(s):  
Reza Naserasr ◽  
Edita Rollova ◽  
Eric Sopena
Keyword(s):  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
International Audience

International audience We conjecture that every signed graph of unbalanced girth 2g, whose underlying graph is bipartite and planar, admits a homomorphism to the signed projective cube of dimension 2g1. Our main result is to show that for a given g, this conjecture is equivalent to the corresponding case (k = 2g) of a conjecture of Seymour claiming that every planar k-regular multigraph with no odd edge-cut of less than k edges is k-edge-colorable. To this end, we exhibit several properties of signed projective cubes and establish a folding lemma for planar even signed graphs.

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