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 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 signless Laplacian spectral determination of the join of regular graphs

2014 ◽  
Vol 06 (04) ◽  
pp. 1450050
Author(s):  
Lizhen Xu ◽  
Changxiang He
Keyword(s):  
Regular Graph ◽  
Spectral Determination ◽  
Regular Graphs ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Signless Laplacian

Let G be an r-regular graph with order n, and G ∨ H be the graph obtained by joining each vertex of G to each vertex of H. In this paper, we prove that G ∨ K2is determined by its signless Laplacian spectrum for r = 1, n - 2. For r = n - 3, we show that G ∨ K2is determined by its signless Laplacian spectrum if and only if the complement of G has no triangles.

scholarly journals Spectral Characterizations of Dumbbell Graphs

10.37236/314 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Jianfeng Wang ◽  
Francesco Belardo ◽  
Qiongxiang Huang ◽  
Enzo M. Li Marzi
Keyword(s):  
Spectral Characterization ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Adjacency Spectrum ◽  
Unique Graph ◽  
Vertex Disjoint ◽  
Spectral Characterizations

A dumbbell graph, denoted by $D_{a,b,c}$, is a bicyclic graph consisting of two vertex-disjoint cycles $C_a$, $C_b$ and a path $P_{c+3}$ ($c \geq -1$) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of $D_{a,b,0}$ (without cycles $C_4$) with $\gcd(a,b)\geq 3$, and we complete the research started in [J.F. Wang et al., A note on the spectral characterization of dumbbell graphs, Linear Algebra Appl. 431 (2009) 1707–1714]. In particular we show that $D_{a,b,0}$ with $3 \leq \gcd(a,b) < a$ or $\gcd(a,b)=a$ and $b\neq 3a$ is determined by the spectrum. For $b=3a$, we determine the unique graph cospectral with $D_{a,3a,0}$. Furthermore we give the spectral characterization w.r.t. the signless Laplacian spectrum of all dumbbell 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.

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. 

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.

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 Research on Extreme Signed Graphs with Minimal Energy in Tricyclic Signed Graphs S(n, n + 2)

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Yajing Wang ◽  
Yubin Gao
Keyword(s):  
Complex Systems ◽  
Computer Models ◽  
Simple Graph ◽  
Signed Graph ◽  
Minimal Energy ◽  
Signed Graphs ◽  
The Absolute

A signed graph is acquired by attaching a sign to each edge of a simple graph, and the signed graphs have been widely used as significant computer models in the study of complex systems. The energy of a signed graph S can be described as the sum of the absolute values of its eigenvalues. In this paper, we characterize tricyclic signed graphs with minimal energy.

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