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.

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.

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.

scholarly journals Bounds on the $A_{\alpha}$-spread of a graph

2020 ◽  
Vol 36 (36) ◽  
pp. 214-227 ◽  
Author(s):  
Zhen Lin ◽  
Lianying Miao ◽  
Shu-Guang Guo
Keyword(s):  
Real Number ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Lower And Upper Bounds ◽  
Largest Eigenvalue ◽  
Smallest Eigenvalue ◽  
The Difference ◽  
The Largest Eigenvalue

Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

scholarly journals General Bounds for Identifying Codes in Some Infinite Regular Graphs

10.37236/1583 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
Irène Charon ◽  
Iiro Honkala ◽  
Olivier Hudry ◽  
Antoine Lobstein
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Regular Graphs ◽  
Lower And Upper Bounds ◽  
Identifying Codes ◽  
Identifying Code

Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.

scholarly journals Harary index of the k-th power of a graph

2013 ◽  
Vol 7 (1) ◽  
pp. 94-105 ◽  
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman
Keyword(s):  
Type Inequality ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph

The k-th power of a graph G, denoted by Gk, is a graph with the same set of vertices as G, such that two vertices are adjacent in Gk if and only if their distance in G is at most k. The Harary index H is the sum of the reciprocal distances of all pairs of vertices of the underlying graph. Lower and upper bounds on H(Gk) are obtained. A Nordhaus-Gaddum type inequality for H(Gk) is also established.

scholarly journals Degree Constrained Orientations in Countable Graphs

10.37236/846 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Attila Bernáth ◽  
Henning Bruhn
Keyword(s):  
Undirected Graph ◽  
Upper Bounds ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lower And Upper Bounds ◽  
Finite Graphs ◽  
Degree Function ◽  
Necessary And Sufficient

Degree constrained orientations are orientations of an (undirected) graph where the in-degree function satisfies given lower and upper bounds. For finite graphs Frank and Gyárfás (1976) gave a necessary and sufficient condition for the existence of such an orientation. We extend their result to countable graphs.

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

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.

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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