Results 11 to 20 of about 15,842 (155)

Fault-tolerant metric dimension of zero-divisor graphs of commutative rings [PDF]

AKCE International Journal of Graphs and Combinatorics, 2021
Let R be a commutative ring with identity. The zero-divisor graph of R denoted by is an undirected graph where is the set of non-zero zero-divisors of R and there is an edge between the vertices z1 and z2 in if A set of vertices S resolves a graph G if ...
Sahil Sharma, Vijay Kumar Bhat
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Fault-Tolerant Metric Dimension in Carbon Networks

Foundations
In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every ...
Kamran Azhar, Asim Nadeem, Yilun Shang
doaj   +2 more sources

Studies of Multilevel Networks via Fault-Tolerant Metric Dimensions

IEEE Access, 2022
A subset $T$ of the vertex set of a network $G$ is called a resolving set for $G$ if each pair of vertices of $G$ have distinct representations with respect to $T$ . A resolving set $B^{\prime} $ among all the resolving sets of a network $G$
Imtiaz Ali, Muhammad Javaid, Samuel Asefa Fufa   +2 more
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Fault-Tolerant Metric Dimension and Applications: Zero-Divisor Graph of Upper Triangular Matrices

Mathematics
Graph invariants play a crucial role in understanding the structural and combinatorial characteristics of graphs. The fault-tolerant metric dimension, as a significant graph invariant, finds applications in diverse areas such as robust network ...
Latif Abdelmalek Hanna, Maryam M. Alkandari, Vijay Kumar Bhat   +2 more
doaj   +2 more sources

Investigating the Metric and Fault-Tolerant Dimensions in Para-Line Network Topologies

AKCE International Journal of Graphs and Combinatorics
The resolving set (RS) and metric dimension (MD) are critical concepts used in various fields such as computer networks, robot navigation, chemical structures, communication networks, transportation, and electric circuits.
M. Faheem, M. Ahmad, Z. Zahid, Njood. S. E. ALmutire, Hamiden Abd El-Wahed Khalifa, Taha Radwan   +5 more
doaj   +2 more sources

Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks [PDF]

Applied Mathematics and Computation, 2022
A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for every $x \in S$.
Prabhu, S., Manimozhi, V., Arulperumjothi, M., Klavžar, Sandi   +3 more
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Fault-Tolerant Edge Metric Dimension of Zero-Divisor Graphs of Commutative Rings

Axioms
In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings.
Omaima Alshanquiti, Malkesh Singh, Vijay Kumar Bhat   +2 more
doaj   +2 more sources

On Fault-Tolerant Resolving Sets of Some Families of Ladder Networks

Complexity, 2021
In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an
Hua Wang, Muhammad Azeem, Muhammad Faisal Nadeem, Ata Ur-Rehman, Adnan Aslam   +4 more
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FAULT-TOLERANT METRIC DIMENSION OF CIRCULANT GRAPHS [PDF]

Facta Universitatis, Series: Mathematics and Informatics, 2019
A set $W$ of vertices in a graph $G$ is called a resolving setfor $G$ if for every pair of distinct vertices $u$ and $v$ of $G$ there exists a vertex $w \in W$ such that the distance between $u$ and $w$ is different from the distance between $v$ and $w$. The cardinality of a minimum resolving set is called the metric dimension of $G$, denoted by $\beta(
Seyedi, Narjes, Maimani, Hamid Reza
openaire   +1 more source

On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs

Mathematics, 2022
Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n,
Sakander Hayat, Asad Khan, Yubin Zhong
doaj   +1 more source