scholarly journals Characterization of 2-Path Product Signed Graphs with Its Properties

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Deepa Sinha ◽  
Deepakshi Sharma
Keyword(s):  
Social Sciences ◽  
Simple Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Positive Edge ◽  
Friendly Relation ◽  
Negative Edge ◽  
Vertex Set

A signed graph is a simple graph where each edge receives a sign positive or negative. Such graphs are mainly used in social sciences where individuals represent vertices friendly relation between them as a positive edge and enmity as a negative edge. In signed graphs, we define these relationships (edges) as of friendship (“+” edge) or hostility (“-” edge). A 2-path product signed graph S#^S of a signed graph S is defined as follows: the vertex set is the same as S and two vertices are adjacent if and only if there exists a path of length two between them in S. The sign of an edge is the product of marks of vertices in S where the mark of vertex u in S is the product of signs of all edges incident to the vertex. In this paper, we give a characterization of 2-path product signed graphs. Also, some other properties such as sign-compatibility and canonically-sign-compatibility of 2-path product signed graphs are discussed along with isomorphism and switching equivalence of this signed graph with 2-path signed graph.

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.

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 Toward a Laplacian spectral determination of signed ∞-graphs

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2283-2294 ◽  
Author(s):  
Mohammad Iranmanesh ◽  
Mahboubeh Saheli
Keyword(s):  
Simple Graph ◽  
Spectral Characterization ◽  
Signed Graph ◽  
Spectral Determination ◽  
Laplacian Spectrum ◽  
Signed Graphs ◽  
Spectral Invariants

A signed graph consists of a (simple) graph G=(V,E) together with a function ? : E ? {+,-} called signature. Matrices can be associated to signed graphs and the question whether a signed graph is determined by the set of its eigenvalues has gathered the attention of several researchers. In this paper we study the spectral determination with respect to the Laplacian spectrum of signed ?-graphs. After computing some spectral invariants and obtain some constraints on the cospectral mates, we obtain some non isomorphic signed graphs cospectral to signed ?-graphs and we study the spectral characterization of the signed ?-graphs containing a triangle.

scholarly journals On the Unitary Cayley Signed Graphs

10.37236/716 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Deepa Sinha ◽  
Pravin Garg
Keyword(s):  
Positive Integer ◽  
Cayley Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Prime Factors ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Sigma E

A $signed graph$ (or $sigraph$ in short) is an ordered pair $S = (S^u, \sigma)$, where $S^u$ is a graph $G = (V, E)$ and $\sigma : E\rightarrow \{+,-\}$ is a function from the edge set $E$ of $S^u$ into the set $\{+, -\}$. For a positive integer $n > 1$, the unitary Cayley graph $X_n$ is the graph whose vertex set is $Z_n$, the integers modulo $n$ and if $U_n$ denotes set of all units of the ring $Z_n$, then two vertices $a, b$ are adjacent if and only if $a-b \in U_n$. For a positive integer $n > 1$, the unitary Cayley sigraph $\mathcal{S}_n = (\mathcal{S}^u_n, \sigma)$ is defined as the sigraph, where $\mathcal{S}^u_n$ is the unitary Cayley graph and for an edge $ab$ of $\mathcal{S}_n$, $$\sigma(ab) = \begin{cases} + & \text{if } a \in U_n \text{ or } b \in U_n,\\ - & \text{otherwise.} \end{cases}$$ In this paper, we have obtained a characterization of balanced unitary Cayley sigraphs. Further, we have established a characterization of canonically consistent unitary Cayley sigraphs $\mathcal{S}_n$, where $n$ has at most two distinct odd prime factors.

scholarly journals Characterization of 2-Path Signed Network

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Deepa Sinha ◽  
Deepakshi Sharma
Keyword(s):  
Communication Networks ◽  
Positive Sign ◽  
Theoretic Model ◽  
Wireless Communication Networks ◽  
Signed Graphs ◽  
Negative Edge ◽  
Vertex Set ◽  
Signed Network ◽  
And Control

A signed network is a network where each edge receives a sign: positive or negative. In this paper, we report our investigation on 2-path signed network of a given signed network Σ , which is defined as the signed network whose vertex set is that of Σ and two vertices in Σ 2 are adjacent if there exist a path of length two between them in Σ . An edge ab in Σ 2 receives a negative sign if all the paths of length two between them are negative, otherwise it receives a positive sign. A signed network is said to be if clusterable its vertex set can be partitioned into pairwise disjoint subsets, called clusters, such that every negative edge joins vertices in different clusters and every positive edge joins vertices in the same clusters. A signed network is balanced if it is clusterable with exactly two clusters. A signed network is sign-regular if the number of positive (negative) edges incident to each vertex is the same for all the vertices. We characterize the 2-path signed graphs as balanced, clusterable, and sign-regular along with their respective algorithms. The 2-path network along with these characterizations is used to develop a theoretic model for the study and control of interference of frequency in wireless communication networks.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

scholarly journals Circular Chromatic Number of Signed Graphs

10.37236/9938 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Reza Naserasr ◽  
Zhouningxin Wang ◽  
Xuding Zhu
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Simple Graph ◽  
Signed Graph ◽  
Signed Graphs ◽  
Outerplanar Graphs ◽  
Circular Chromatic Number ◽  
Basic Properties ◽  
Large Girth ◽  
Sigma E

A signed graph is a pair $(G, \sigma)$, where $G$ is a graph (loops and multi edges allowed) and $\sigma: E(G) \to \{+, -\}$ is a signature which assigns to each edge of $G$ a sign. Various notions of coloring of signed graphs have been studied. In this paper, we extend circular coloring of graphs to signed graphs. Given a signed graph $(G, \sigma)$ with no positive loop, a circular $r$-coloring of $(G, \sigma)$ is an assignment $\psi$ of points of a circle of circumference $r$ to the vertices of $G$ such that for every edge $e=uv$ of $G$, if $\sigma(e)=+$, then $\psi(u)$ and $\psi(v)$ have distance at least $1$, and if $\sigma(e)=-$, then $\psi(v)$ and the antipodal of $\psi(u)$ have distance at least $1$. The circular chromatic number $\chi_c(G, \sigma)$ of a signed graph $(G, \sigma)$ is the infimum of those $r$ for which $(G, \sigma)$ admits a circular $r$-coloring. For a graph $G$, we define the signed circular chromatic number of $G$ to be $\max\{\chi_c(G, \sigma): \sigma \text{ is a signature of $G$}\}$.  We study basic properties of circular coloring of signed graphs and develop tools for calculating $\chi_c(G, \sigma)$. We explore the relation between the circular chromatic number and the signed circular chromatic number of graphs, and present bounds for the signed circular chromatic number of some families of graphs. In particular,  we determine the supremum of the signed circular chromatic number of $k$-chromatic graphs of large girth, of simple bipartite planar graphs, $d$-degenerate graphs, simple outerplanar graphs and series-parallel graphs. We construct a signed planar simple graph whose circular chromatic number is $4+\frac{2}{3}$. This is based and improves on a signed graph built by Kardos and Narboni as a counterexample to a conjecture of Máčajová, Raspaud, and Škoviera. 

Clique doubly connected domination in the join and lexicographic product of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050066
Author(s):  
Enrico L. Enriquez ◽  
Albert D. Ngujo
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Connected Dominating Set ◽  
Lexicographic Product ◽  
Connected Domination Number ◽  
Vertex Set ◽  
Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two 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.

BALANCED UNITARY CAYLEY SIGRAPHS OVER FINITE COMMUTATIVE RINGS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350152 ◽  
Author(s):  
YOTSANAN MEEMARK ◽  
BORWORN SUNTORNPOCH
Keyword(s):  
Cayley Graph ◽  
Line Graph ◽  
Commutative Rings ◽  
Signed Graph ◽  
Group Of Units ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Finite Commutative Rings

Let R be a finite commutative ring with identity 1. The unitary Cayley graph of R, denoted by GR, is the graph whose vertex set is R and the edge set {{a, b} : a, b ∈ R and a - b ∈ R×}, where R× is the group of units of R. We define the unitary Cayley signed graph (or unitary Cayley sigraph in short) to be an ordered pair 𝒮R = (GR, σ), where GR is the unitary Cayley graph over R with signature σ : E(GR) → {1, -1} given by [Formula: see text] In this paper, we give a criterion on R for SR to be balanced (every cycle in 𝒮R is positive) and a criterion for its line graph L(𝒮R) to be balanced. We characterize all finite commutative rings with the property that the marked sigraph 𝒮R,μ is canonically consistent. Moreover, we give a characterization of all finite commutative rings where 𝒮R, η(𝒮R) and L(𝒮R) are hyperenergetic balanced.

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