scholarly journals Radial Radio Number of Hexagonal and Its Derived Networks

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Kins Yenoke ◽  
Mohammed K. A. Kaabar ◽  
Mohammed M. Ali Al-Shamiri ◽  
R. C. Thivyarathi
Keyword(s):  
Connected Graph ◽  
Radio Number ◽  
Radio Labelling

A mapping   ℸ :   V G ⟶ N ∪ 0 for a connected graph G = V , E is called a radial radio labelling if it satisfies the inequality     ℸ x −   ℸ   y + d x , y ≥ rad G + 1 ∀ x , y ∈ V G , where rad G is the radius of the graph G . The radial radio number of ℸ denoted by r r ℸ   is the maximum number mapped under ℸ . The radial radio number of G denoted by r r G is equal to min { r r ℸ   / ℸ   is a radial radio labelling of G }.

scholarly journals Zagreb Radio Indices and Coindices

2021 ◽  
Vol 11 (1) ◽  
Keyword(s):  
Radio Number ◽  
Radio Labelling

Zagreb radio index or coindex of a graph is defined based on the optimal radio labelling of the graph, where optimal refers to the labeling with a span equal to the radio number of the graph. We have defined the first and second Zagreb radio coindex, third Zagreb radio index, and coindex and calculated its value for some famous graphs. Furthermore, we established a certain relationship between Zagreb radio indices and the coindices of graphs.

scholarly journals The Radial Radio Number and the Clique Number of a Graph

2019 ◽  
Vol 9 (1S4) ◽  
pp. 1002-1006
Keyword(s):  
Connected Graph ◽  
Clique Number ◽  
Negative Integer ◽  
Radio Number ◽  
The Difference ◽  
The Given ◽  
Positive Integers ◽  
Radio Labeling

Let G(V(G),E(G)) be a graph. A radial radio labeling, f, of a connected graph G is an assignment of positive integers to the vertices satisfying the following condition: d(u, v) | f (u)  f (v) | 1  r(G) , for any two distinct vertices u, v V(G) , where d(u,v) and r(G) denote the distance between the vertices u and v and the radius of the graph G, respectively. The span of a radial radio labeling f is the largest integer in the range of f and is denoted by span(f). The radial radio number of G, r(G) , is the minimum span taken over all radial radio labelingsof G. In this paper, we construct a graph a graph for which the difference between the radial radio number and the clique number is the given non negative integer.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Privacy Preserving Distributed Summation in a Connected Graph

2020 ◽  
Vol 53 (2) ◽  
pp. 3445-3450
Author(s):  
Katrine Tjell ◽  
Rafael Wisniewski
Keyword(s):  
Connected Graph ◽  
Privacy Preserving

scholarly journals The normalized distance Laplacian

2021 ◽  
Vol 9 (1) ◽  
pp. 1-18
Author(s):  
Carolyn Reinhart
Keyword(s):  
Spectral Radius ◽  
Characteristic Polynomial ◽  
Adjacency Matrix ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Diagonal Matrix ◽  
The Matrix

Abstract The distance matrix 𝒟(G) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T(G) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟𝒧(G) = I−T(G)−1/2 𝒟(G)T(G)−1/2, is introduced. This is analogous to the normalized Laplacian matrix, 𝒧(G) = I − D(G)−1/2 A(G)D(G)−1/2, where D(G) is the diagonal matrix of degrees of the vertices of the graph and A(G) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟𝒧-cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

scholarly journals On minimum algebraic connectivity of graphs whose complements are bicyclic

2019 ◽  
Vol 17 (1) ◽  
pp. 1490-1502 ◽  
Author(s):  
Jia-Bao Liu ◽  
Muhammad Javaid ◽  
Mohsin Raza ◽  
Naeem Saleem
Keyword(s):  
Connected Graph ◽  
Diffusion Processes ◽  
Laplacian Matrix ◽  
Group Differences ◽  
Algebraic Connectivity ◽  
Connected Graphs ◽  
Bicyclic Graph ◽  
Smallest Eigenvalue ◽  
Unique Graph ◽  
The Stability

Abstract The second smallest eigenvalue of the Laplacian matrix of a graph (network) is called its algebraic connectivity which is used to diagnose Alzheimer’s disease, distinguish the group differences, measure the robustness, construct multiplex model, synchronize the stability, analyze the diffusion processes and find the connectivity of the graphs (networks). A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. In this paper, firstly the unique graph with a minimum algebraic connectivity is characterized in the class of connected graphs whose complements are bicyclic with exactly three cycles. Then, we find the unique graph of minimum algebraic connectivity in the class of connected graphs $\begin{array}{} {\it\Omega}^c_{n}={\it\Omega}^c_{1,n}\cup{\it\Omega}^c_{2,n}, \end{array}$ where $\begin{array}{} {\it\Omega}^c_{1,n} \end{array}$ and $\begin{array}{} {\it\Omega}^c_{2,n} \end{array}$ are classes of the connected graphs in which the complement of each graph of order n is a bicyclic graph with exactly two and three cycles, respectively.

The Maximum Genus on a 3-Vertex-Connected Graph

2000 ◽  
Vol 16 (2) ◽  
pp. 159-164 ◽  
Author(s):  
Yuangqiu Huang
Keyword(s):  
Connected Graph ◽  
Maximum Genus

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

scholarly journals Data-driven graph drawing techniques with applications for conveyor systems

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Simone Göttlich ◽  
Sven Spieckermann ◽  
Stephan Stauber ◽  
Andrea Storck
Keyword(s):  
Real World ◽  
Connected Graph ◽  
Stress Function ◽  
Graph Drawing ◽  
Point Of View ◽  
Data Driven ◽  
Challenging Problem ◽  
Conveyor System ◽  
System Graph ◽  
Real World Problems

AbstractThe visualization of conveyor systems in the sense of a connected graph is a challenging problem. Starting from communication data provided by the IT system, graph drawing techniques are applied to generate an appealing layout of the conveyor system. From a mathematical point of view, the key idea is to use the concept of stress majorization to minimize a stress function over the positions of the nodes in the graph. Different to the already existing literature, we have to take care of special features inspired by the real-world problems.

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