scholarly journals A decomposition of signed graphs with two eigenvalues

Filomat ◽  
2020 ◽  
Vol 34 (6) ◽  
pp. 1949-1957
Author(s):  
Zoran Stanic
Keyword(s):  
Set Partition ◽  
Signed Graphs ◽  
Vertex Set ◽  
Spectral Characterizations ◽  
Theoretical Results

In this study we consider connected signed graphs with 2 eigenvalues that admit a vertex set partition such that the induced signed graphs also have 2 eigenvalues, each. We derive some spectral characterizations, along with examples supported by additional theoretical results. We also prove an inequality that is a fundamental ingredient for the resolution of the Sensitivity Conjecture.

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.

2K2 vertex-set partition into nonempty parts

2008 ◽  
Vol 30 ◽  
pp. 291-296 ◽  
Author(s):  
Simone Dantas ◽  
Elaine M. Eschen ◽  
Luerbio Faria ◽  
Celina M.H. de Figueiredo ◽  
Sulamita Klein
Keyword(s):  
Set Partition ◽  
Vertex Set

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 2K2 vertex-set partition into nonempty parts

2010 ◽  
Vol 310 (6-7) ◽  
pp. 1259-1264 ◽  
Author(s):  
Kathryn Cook ◽  
Simone Dantas ◽  
Elaine M. Eschen ◽  
Luerbio Faria ◽  
Celina M.H. de Figueiredo ◽  
...  
Keyword(s):  
Set Partition ◽  
Vertex Set

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.

scholarly journals Signed graphs connected with the root lattice

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 157-160
Author(s):  
RN Yadav
Keyword(s):  
Connected Graph ◽  
Signed Graph ◽  
Connected Components ◽  
Root Lattice ◽  
Signed Graphs ◽  
Cut Vertex ◽  
Vertex Set

For any base of the root lattice (An) we can construct a signed graph. A signed graph is one whose edges are signed by +1 or -1. A signed graph is balanced if and only if its vertex set can be divided into two sets-either of which may be empty–so that each edge between the sets is negative and each edge within a set is positive. For a given signed graph Tsaranov, Siedel and Cameron constructed the corresponding root lattice. In the present work we have dealt with signed graphs corresponding to the root lattice An. A connected graph is called a Fushimi tree if its all blocks are complete subgraphs. A Fushimi tree is said to be simple when by deleting any cut vertex we have always two connected components. A signed Fushimi tree is called a Fushimi tree with standard sign if it can be transformed into a signed Fushimi tree whose all edges are signed by +1 by switching. Here we have proved that any signed graph corresponding to An is a simple Fushimi tree with standard sign. Our main result is that s simple Fushimi tree with standard sign is contained in the cluster given by a line. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10396 BIBECHANA 11(1) (2014) 157-160

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.

scholarly journals Counting Connected Set Partitions of Graphs

10.37236/501 ◽  
2010 ◽  
Vol 18 (1) ◽  
Author(s):  
Frank Simon ◽  
Peter Tittmann ◽  
Martin Trinks
Keyword(s):  
Undirected Graph ◽  
Set Partition ◽  
Set Partitions ◽  
Connected Set ◽  
Vertex Set

Let $G=(V,E)$ be a simple undirected graph with $n$ vertices then a set partition $\pi=\{V_1, \ldots, V_k\}$ of the vertex set of $G$ is a connected set partition if each subgraph $G[V_j]$ induced by the blocks $V_j$ of $\pi$ is connected for $1\le j\le k$. Define $q_{i}(G)$ as the number of connected set partitions in $G$ with $i$ blocks. The partition polynomial is $Q(G, x)=\sum_{i=0}^n q_{i}(G)x^i$. This paper presents a splitting approach to the partition polynomial on a separating vertex set $X$ in $G$ and summarizes some properties of the bond lattice. Furthermore the bivariate partition polynomial $Q(G,x,y)=\sum_{i=1}^n \sum_{j=1}^m q_{ij}(G)x^iy^j$ is briefly discussed, where $q_{ij}(G)$ counts the number of connected set partitions with $i$ blocks and $j$ intra block edges. Finally the complexity for the bivariate partition polynomial is proven to be $\sharp P$-hard.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

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