scholarly journals On a Question of Prime Labeling of Graphs

2021 ◽  
pp. 87-93
Author(s):  
A. M. C. U. M. Athapattu ◽  
P. G. R. S. Ranasinghe
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Undirected Graph ◽  
Prime Labeling

In the field of graph theory, the complete graph  of  vertices is a simple undirected graph such that every pair of distinct vertices is connected by a unique edge. In the present work, we introduce planar subgraph  of  with maximal number of edges . Generally,  does not admit prime labeling. We present an algorithm to obtain prime-labeled subgraphs of  . We conclude the paper by stating two conjectures based on labeling of . In particular, the planar subgraph admits anti-magic labeling but does not admit edge magic total labeling.

scholarly journals Related graphs of the conjugacy classes of a 3-generator 5-group

2018 ◽  
Vol 14 ◽  
pp. 454-456
Author(s):  
Alia Husna Mohd Noor ◽  
Nor Haniza Sarmin ◽  
Hamisan Rahmat
Keyword(s):  
Graph Theory ◽  
Conjugacy Class ◽  
Group Action ◽  
Equivalence Relation ◽  
Complete Graph ◽  
Undirected Graph ◽  
Conjugacy Classes ◽  
Group Presentation ◽  
Class Graph

The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of  which is of size two where  and  commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.

scholarly journals The Study of the Theoretical Size and Node Probability of the Loop Cutset in Bayesian Networks

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1079 ◽  
Author(s):  
Jie Wei ◽  
Yufeng Nie ◽  
Wenxian Xie
Keyword(s):  
Probability Theory ◽  
Graph Theory ◽  
Bayesian Inference ◽  
Bayesian Networks ◽  
Numerical Simulations ◽  
Complete Graph ◽  
Upper Bound ◽  
Numerical Algorithms ◽  
Theoretical Research ◽  
Theoretical Results

Pearl’s conditioning method is one of the basic algorithms of Bayesian inference, and the loop cutset is crucial for the implementation of conditioning. There are many numerical algorithms for solving the loop cutset, but theoretical research on the characteristics of the loop cutset is lacking. In this paper, theoretical insights into the size and node probability of the loop cutset are obtained based on graph theory and probability theory. It is proven that when the loop cutset in a p-complete graph has a size of p − 2 , the upper bound of the size can be determined by the number of nodes. Furthermore, the probability that a node belongs to the loop cutset is proven to be positively correlated with its degree. Numerical simulations show that the application of the theoretical results can facilitate the prediction and verification of the loop cutset problem. This work is helpful in evaluating the performance of Bayesian networks.

A ZERO-DIVISOR GRAPH FOR MODULES WITH RESPECT TO THEIR (FIRST) DUAL

2012 ◽  
Vol 12 (02) ◽  
pp. 1250151 ◽  
Author(s):  
M. BAZIAR ◽  
E. MOMTAHAN ◽  
S. SAFAEEYAN
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Complete Graph ◽  
Projective Module ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Clique Number ◽  
Covariant Functor ◽  
Zero Divisor Graph

Let M be an R-module. We associate an undirected graph Γ(M) to M in which nonzero elements x and y of M are adjacent provided that xf(y) = 0 or yg(x) = 0 for some nonzero R-homomorphisms f, g ∈ Hom (M, R). We observe that over a commutative ring R, Γ(M) is connected and diam (Γ(M)) ≤ 3. Moreover, if Γ(M) contains a cycle, then gr (Γ(M)) ≤ 4. Furthermore if ∣Γ(M)∣ ≥ 1, then Γ(M) is finite if and only if M is finite. Also if Γ(M) = ∅, then any nonzero f ∈ Hom (M, R) is monic (the converse is true if R is a domain). For a nonfinitely generated projective module P we observe that Γ(P) is a complete graph. We prove that for a domain R the chromatic number and the clique number of Γ(M) are equal. When R is self-injective, we will also observe that the above adjacency defines a covariant functor between a subcategory of R-MOD and the Category of graphs.

Binomial incidence matrix of a semigraph

2020 ◽  
pp. 2150017
Author(s):  
Jyoti Shetty ◽  
G. Sudhakara
Keyword(s):  
Graph Theory ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Incidence Matrix ◽  
Systematic Approach ◽  
Line Graph ◽  
The Matrix ◽  
Theory Of Graphs

A semigraph, defined as a generalization of graph by  Sampathkumar, allows an edge to have more than two vertices. The idea of multiple vertices on edges gives rise to multiplicity in every concept in the theory of graphs when generalized to semigraphs. In this paper, we define a representing matrix of a semigraph [Formula: see text] and call it binomial incidence matrix of the semigraph [Formula: see text]. This matrix, which becomes the well-known incidence matrix when the semigraph is a graph, represents the semigraph uniquely, up to isomorphism. We characterize this matrix and derive some results on the rank of the matrix. We also show that a matrix derived from the binomial incidence matrix satisfies a result in graph theory which relates incidence matrix of a graph and adjacency matrix of its line graph. We extend the concept of “twin vertices” in the theory of graphs to semigraph theory, and characterize them. Finally, we derive a systematic approach to show that the binomial incidence matrix of any semigraph on [Formula: see text] vertices can be obtained from the incidence matrix of the complete graph [Formula: see text].

Energy and skew-energy of a modified graph

2021 ◽  
Vol 30 (1) ◽  
pp. 41-48
Author(s):  
V. LOKESHA ◽  
Y. SHANTHAKUMARI ◽  
K. ZEBA YASMEEN
Keyword(s):  
Complete Graph ◽  
Scientific Community ◽  
Undirected Graph ◽  
Molecular Chemistry ◽  
Graph Operations ◽  
Skew Energy ◽  
Direct Applicability

Graph energies draw the greater attention of the scientific community due to their direct applicability in molecular chemistry. In this paper, we establish the energy of a graph obtained by the means of some graph operations. The energy of the product of graph Kn × G, where Kn is a complete graph and G is a simple undirected graph and energy of the corresponding digraph are estimated. Further, the duplication graph DG is considered and proved that the energy E(DG) = 2E(G) and E(DGσ) = 2E(Gσ).

6. Graphs

2016 ◽  
pp. 86-108
Author(s):  
Robin Wilson
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Travelling Salesman Problem ◽  
Chemical Bonds ◽  
Hamiltonian Graphs ◽  
Road Map ◽  
Cycle Graph

Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits the Königsberg bridges problem, the knight’s tour problem, the Gas–Water–Electricity problem, the map-colour problem, the minimum connector problem, and the travelling salesman problem and explains how they can all be considered as problems in graph theory. It begins with an explanation of a graph and describes the complete graph, the complete bipartite graph, and the cycle graph, which are all simple graphs. It goes on to describe trees in graph theory, Eulerian and Hamiltonian graphs, and planar graphs.

scholarly journals On SD-Prime Labelings of Some One Point Unions of Gears

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 849
Author(s):  
Wai-Chee Shiu ◽  
Gee-Choon Lau
Keyword(s):  
Undirected Graph ◽  
Prime Labeling ◽  
Edge Labeling

Let G=(V(G),E(G)) be a simple, finite and undirected graph of order n. Given a bijection f:V(G)→{1,…,n}, and every edge uv in E(G), let S=f(u)+f(v) and D=|f(u)−f(v)|. The labeling f induces an edge labeling f′:E(G)→{0,1} such that for an edge uv in E(G), f′(uv)=1 if gcd(S,D)=1, and f′(uv)=0 otherwise. Such a labeling is called an SD-prime labeling if f′(uv)=1 for all uv∈E(G). We provide SD-prime labelings for some one point unions of gear graphs.

scholarly journals PELABELAN TOTAL TAK-AJAIB SISI PADA MULTISTAR

2021 ◽  
Vol 14 (2) ◽  
pp. 62
Author(s):  
Arif Budi Prasetyo ◽  
Yoga Jati Kusuma
Keyword(s):  
Graph Theory ◽  
Research Method ◽  
Undirected Graph ◽  
Graph Labeling ◽  
Star Graphs ◽  
Literature Research ◽  
The Difference

Graph theory contains several topics, one of which will be discussed in this study is graph labeling. In the topic of labeling, the graph used is a limited, simple, and undirected graph. In this study, the type of labeling used is total labeling. The multistar graph used in this study is a combination of star graphs whose center vertex are not connected to each other. This research uses literature research method which is divided into two parts, that is the basic calculation to determine the boundary of the first term a and the difference d from the (a,d) edge antimagic total labeling on the mS_n multistar graph. The second part is to apply (a,d) edge antimagic total labeling  to the multistar graph mS_n. Further result shown on conclusion. Teori graf memuat beberapa topik, salah satu yang akan dibahas dalam penelitian ini adalah pelabelan graf. Dalam topik pelabelan, graf yang digunakan merupakan graf terbatas, sederhana, dan tak berarah. Dalam penelitian ini, jenis pelabelan yang digunakan adalah pelabelan total. Graf multistar yang digunakan dalam penelitian ini merupakan gabungan graf star yang titik pusatnya tidak saling terhubung. Penelitian ini menggunakan metode penelitian pustaka yang dibagi menjadi dua bagian, yakni perhitungan dasar untuk menentukan batas suku pertama a dan beda d dari pelabelan total tak-ajaib sisi (a,d) pada graf multistar mS_n. Bagian yang kedua adalah keberlakukan pelabelan total tak-ajaib sisi (a,d) pada graf multistar mS_n. Hasil lebih lanjut dijelaskan pada bagian kesimpulan.

scholarly journals Signless Laplacian determinations of some graphs with independent edges

2018 ◽  
Vol 10 (1) ◽  
pp. 185-196 ◽  
Author(s):  
R. Sharafdini ◽  
A.Z. Abdian
Keyword(s):  
Natural Number ◽  
Complete Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Laplacian Spectrum ◽  
Signless Laplacian Spectrum ◽  
Laplacian Spectra ◽  
Signless Laplacian ◽  
Degree Matrix

Let $G$ be a simple undirected graph. Then the signless Laplacian matrix of $G$ is defined as $D_G + A_G$ in which $D_G$ and $A_G$ denote the degree matrix and the adjacency matrix of $G$, respectively. The graph $G$ is said to be determined by its signless Laplacian spectrum (DQS, for short), if any graph having the same signless Laplacian spectrum as $G$ is isomorphic to $G$. We show that $G\sqcup rK_2$ is determined by its signless Laplacian spectra under certain conditions, where $r$ and $K_2$ denote a natural number and the complete graph on two vertices, respectively. Applying these results, some DQS graphs with independent edges are obtained.

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