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29 pages, 352 KB

Open AccessArticle

The Euler Sombor Index and the Average Sombor Index of the Cozero-Divisor Graph over the Ring

Z_{n}

by Amal S. Alali, Muzibur Rahman Mozumder and Mohd Salman Ali

Mathematics 2026, 14(3), 414; https://doi.org/10.3390/math14030414 - 25 Jan 2026

Viewed by 72

Abstract

Let

Z {(R)}^{'}

denote the set of all elements in the ring

R

that are neither zero nor units, where

R

is assumed to be a commutative ring with a multiplicative identity satisfying

1 \neq 0 .

Two distinct vertices [...] Read more.

Let

Z {(R)}^{'}

denote the set of all elements in the ring

R

that are neither zero nor units, where

R

is assumed to be a commutative ring with a multiplicative identity satisfying

1 \neq 0 .

Two distinct vertices w and

κ

are defined to be adjacent if and only if

κ

does not lie in the ideal generated by w in

R,

that is,

κ \notin w R,

and simultaneously, w does not lie in the ideal generated by

κ

in

R,

that is,

w \notin κ R .

The cozero-divisor graph of

R,

denoted by

Γ^{'} (R),

is an undirected graph in which the vertices are given by the set

Z {(R)}^{'} .

This article presents a comprehensive evaluation of both the Euler Sombor index and the average Sombor index for the graphs

Γ^{'} (Z_{n})

corresponding to various values of

n .

Full article

12 pages, 414 KB

Open AccessArticle

Fault-Tolerant Metric Dimension and Applications: Zero-Divisor Graph of Upper Triangular Matrices

by Latif Abdelmalek Hanna, Maryam M. Alkandari and Vijay Kumar Bhat

Mathematics 2025, 13(22), 3678; https://doi.org/10.3390/math13223678 - 17 Nov 2025

Viewed by 348

Abstract

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 optimization, autonomous robot navigation and intelligent sensor systems. In this [...] Read more.

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 optimization, autonomous robot navigation and intelligent sensor systems. In this paper, we investigate the fault-tolerant metric dimension and fault-tolerant edge metric dimension of zero-divisor graphs arising from upper triangular matrices over a finite commutative ring. The obtained results contribute to the understanding of metric-based fault tolerance in algebraically structured graphs. Full article

(This article belongs to the Special Issue Applications of Algebra, Geometry, and Optimization to Control Theory, 2nd Edition)

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19 pages, 319 KB

Open AccessArticle

Eigenvalue Characterizations for the Signless Laplacian Spectrum of Weakly Zero-Divisor Graphs on

Z_{n}

by Nazim, Alaa Altassan and Nof T. Alharbi

Mathematics 2025, 13(16), 2689; https://doi.org/10.3390/math13162689 - 21 Aug 2025

Viewed by 701

Abstract

Let

R

be a commutative ring with identity

1 \neq 0

. The weakly zero-divisor graph of

R

, denoted

W_{Γ} (R)

, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of

R

, where [...] Read more.

Let

R

be a commutative ring with identity

1 \neq 0

. The weakly zero-divisor graph of

R

, denoted

W_{Γ} (R)

, is the simple undirected graph whose vertex set consists of the nonzero zero-divisors of

R

, where two distinct vertices

a

and

b

are adjacent if and only if there exist

r \in ann (a)

and

s \in ann (b)

such that

r s = 0

. In this paper, we study the signless Laplacian spectrum of

W_{Γ} (Z_{n})

for several composite forms of n, including

n = p^{2} q^{2}

,

n = p^{2} q r

,

n = p^{m} q^{m}

and

n = p^{m} q r

, where

p, q, r

are distinct primes and

m \geq 2

. By using generalized join decomposition and quotient matrix methods, we obtain explicit eigenvalue formulas for each case, along with structural bounds, spectral integrality conditions and Nordhaus–Gaddum-type inequalities. Illustrative examples with computed spectra are provided to validate the theoretical results, demonstrating the interplay between the algebraic structure of

Z_{n}

and the spectral properties of its weakly zero-divisor graph. Full article

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13 pages, 295 KB

Open AccessArticle

On D_α-Spectrum of the Weakly Zero-Divisor Graph of ℤ_n

by Amal S. Alali, Mohd Rashid, Asif Imtiyaz Ahmad Khan and Muzibur Rahman Mozumder

Mathematics 2025, 13(15), 2385; https://doi.org/10.3390/math13152385 - 24 Jul 2025

Viewed by 593

Abstract

Let us consider the finite commutative ring R, whose unity is

1 \neq 0 .

Its weakly zero-divisor graph, represented as

W Γ (R)

, is a basic undirected graph with two distinct vertices,

c_{1}

and

c_{2}

, [...] Read more.

Let us consider the finite commutative ring R, whose unity is

1 \neq 0 .

Its weakly zero-divisor graph, represented as

W Γ (R)

, is a basic undirected graph with two distinct vertices,

c_{1}

and

c_{2}

, that are adjacent if and only if there exist

r \in

ann (c_{1})

and

s \in

ann (c_{2})

that satisfy the condition

r s = 0

. Let

D (G)

be the distance matrix and

T r (G)

be the diagonal matrix of the vertex transmissions in basic undirected connected graph G. The

D_{α}

matrix of graph G is defined as

D_{α} (G) = α T r (G) + (1 - α) D (G)

for

α \in [0, 1]

. This article finds the

D_{α}

spectrum for the graph

W Γ (Z_{n})

for various values of n and also shows that

W Γ (Z_{n})

for

n = ϑ_{1} ϑ_{2} ϑ_{3} \dots ϑ_{t} η_{1}^{d_{1}} η_{2}^{d_{2}} \dots η_{s}^{d_{s}} (d_{i} \geq 2, t \geq 1, s \geq 0)

, where

ϑ_{i}

’s and

η_{i}

’s are the distinct primes, is

D_{α}

integral. Full article

(This article belongs to the Section E: Applied Mathematics)

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12 pages, 918 KB

Open AccessArticle

Fault-Tolerant Edge Metric Dimension of Zero-Divisor Graphs of Commutative Rings

by Omaima Alshanquiti, Malkesh Singh and Vijay Kumar Bhat

Axioms 2025, 14(7), 499; https://doi.org/10.3390/axioms14070499 - 26 Jun 2025

Viewed by 1014

Abstract

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. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring [...] Read more.

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. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring S with unity. Consider a graph Γ(S) with vertex set V(Γ) = Z*(S), and two vertices in Γ(S) are adjacent if and only if their product is zero. This graph Γ(S) is known as zero-divisor graph of S. Zero-divisor graphs provide a powerful bridge between abstract algebra and graph theory. The zero-divisor graphs for finite commutative rings and their minimum fault-tolerant edge-resolving sets are studied in this article. Through analytical and constructive techniques, we highlight how the algebraic properties of the ring influence the edge metric structure of its associated graph. In addition to this, the existence of a connected graph G having a resolving set of cardinality of 2n + 2 from a star graph

K_{1,2 n}

, is studied. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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13 pages, 280 KB

Open AccessArticle

Exploring Geometrical Properties of Annihilator Intersection Graph of Commutative Rings

by Ali Al Khabyah and Moin A. Ansari

Axioms 2025, 14(5), 336; https://doi.org/10.3390/axioms14050336 - 27 Apr 2025

Cited by 1 | Viewed by 938

Abstract

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not [...] Read more.

Let

Λ

denote a commutative ring with unity and

D (Λ)

denote a collection of all annihilating ideals from

Λ

. An annihilator intersection graph of

Λ

is represented by the notation

AIG (Λ)

. This graph is not directed in nature, where the vertex set is represented by

D {(Λ)}^{*}

. There is a connection in the form of an edge between two distinct vertices

ς

and

ϱ

in

AIG (Λ)

iff

A n n (ς ϱ) \neq A n n (ς) \cap A n n (ϱ)

. In this work, we begin by categorizing commutative rings

Λ

, which are finite in structure, so that

AIG (Λ)

forms a star graph/2-outerplanar graph, and we identify the inner vertex number of

AIG (Λ)

. In addition, a classification of the finite rings where the genus of

AIG (Λ)

is 2, meaning

AIG (Λ)

is a double-toroidal graph, is also investigated. Further, we determine

Λ

, having a crosscap 1 of

AIG (Λ),

indicating that

AIG (Λ)

is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two. Full article

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21 pages, 508 KB

Open AccessArticle

On Zero-Divisor Graphs of Z_n When n Is Square-Free

by Kholood Alnefaie, Nanggom Gammi, Saifur Rahman and Shakir Ali

Axioms 2025, 14(3), 180; https://doi.org/10.3390/axioms14030180 - 28 Feb 2025

Viewed by 3080

Abstract

In this article, some properties of the zero-divisor graph

Γ (Z_{n})

are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph

Γ (Z_{n})

of ring

Z_{n}

is a [...] Read more.

In this article, some properties of the zero-divisor graph

Γ (Z_{n})

are investigated when n is a square-free positive integer. It is shown that the zero-divisor graph

Γ (Z_{n})

of ring

Z_{n}

is a

(2^{k} - 2)

-partite graph when the prime decomposition of n contains k distinct square-free primes using the method of congruence relation. We present some examples, accompanied by graphic representations, to achieve the desired results. It is also obtained that the zero-divisor graph

Γ (Z_{n})

is Eulerian if n is a square-free odd integer. Since

Z_{n}

is a semisimple ring when n is square-free, the results can be generalized to characterize semisimple rings and modules, as well as rings satisfying Artinian and Noetherian conditions through the properties of their zero-divisor graphs. We endeavored to show that

Γ (R)

is a partite graph with a certain condition on n and also that

Γ (R)

is a complete graph when

n = p^{2}

for a prime p as part of a corollary. To prove these results, we employed the assistance of several theoretic congruence relations that grabbed our attention, making the investigation more interesting. Full article

(This article belongs to the Section Algebra and Number Theory)

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13 pages, 279 KB

Open AccessArticle

Exploring the Embedding of the Extended Zero-Divisor Graph of Commutative Rings

by Ali Al Khabyah and Moin A. Ansari

Axioms 2025, 14(3), 170; https://doi.org/10.3390/axioms14030170 - 26 Feb 2025

Cited by 1 | Viewed by 812

Abstract

R_{c}

represents commutative rings that have unity elements. The collection of all zero-divisor elements in

R_{c}

are represented by

D (R_{c})

. We denote an extended zero-divisor graph by the notation

ℸ^{'} (R_{c})

of [...] Read more.

R_{c}

represents commutative rings that have unity elements. The collection of all zero-divisor elements in

R_{c}

are represented by

D (R_{c})

. We denote an extended zero-divisor graph by the notation

ℸ^{'} (R_{c})

of

R_{c} .

This graph has a set of vertices in

D {(R_{c})}^{*}

. The graph

ℸ^{'} (R_{c})

illustrates interactions among the zero-divisor elements of

R_{c} .

Specifically, two different vertices u and y are connected in

ℸ^{'} (R_{c})

iff

u R_{c} \cap Ann (y)

is non-null or

y R_{c} \cap Ann (u)

is non-null. The main idea for this work is to systematically analyze the ring

R_{c}

which is finite for the unique aspect of their extended zero-divisor graph. This study particularly focuses on instances where the extended zero-divisor graph has a genus or crosscap of two. Furthermore, this work aims to thoroughly characterize finite ring

R_{c}

wherein the extended zero-divisor graph

ℸ^{'} (R_{c})

has an outerplanarity index of two. Finally, we determine the book thickness of

ℸ^{'} (R_{c})

for genus at most one. Full article

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14 pages, 302 KB

Open AccessArticle

On the Normalized Laplacian Spectrum of the Zero-Divisor Graph of the Commutative Ring

Z_{p_{1}^{T_{1}} p_{2}^{T_{2}}}

by Ali Al Khabyah, Nazim and Nadeem Ur Rehman

Axioms 2025, 14(1), 37; https://doi.org/10.3390/axioms14010037 - 4 Jan 2025

Cited by 1 | Viewed by 1356

Abstract

The zero-divisor graph of a commutative ring

R

with a nonzero identity, denoted by

Γ (R)

, is an undirected graph where the vertex set

Z {(R)}^{*}

consists of all nonzero zero-divisors of

R

. Two distinct vertices [...] Read more.

The zero-divisor graph of a commutative ring

R

with a nonzero identity, denoted by

Γ (R)

, is an undirected graph where the vertex set

Z {(R)}^{*}

consists of all nonzero zero-divisors of

R

. Two distinct vertices a and b in

Γ (R)

are adjacent if and only if

a b = 0

. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of

Γ (Z_{n})

for specific values of

n

in the set

{p q, p^{2} q, p^{3}, p^{4}}

, where p and q are distinct primes satisfying

p < q

. Motivated by their work, this article investigates the normalized Laplacian spectrum of

Γ (Z_{n})

for a more general class of

n

, where

n

is represented as

p_{1}^{T_{1}} p_{2}^{T_{2}}

, with

p_{1}

and

p_{2}

being distinct primes

(p_{1} < p_{2})

, and

T_{1}, T_{2}

are positive integers. Full article

10 pages, 1215 KB

Open AccessArticle

Exploring Ring Structures: Multiset Dimension Analysis in Compressed Zero-Divisor Graphs

by Nasir Ali, Hafiz Muhammad Afzal Siddiqui, Muhammad Imran Qureshi, Suhad Ali Osman Abdallah, Albandary Almahri, Jihad Asad and Ali Akgül

Symmetry 2024, 16(7), 930; https://doi.org/10.3390/sym16070930 - 20 Jul 2024

Cited by 5 | Viewed by 1685

Abstract

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring

R

and the associated compressed zero-divisor graph. An undirected graph consisting of [...] Read more.

This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring

R

and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set

Z (R_{E}) \ {[0]} = R_{E} \ {[0], [1]}

, where

R_{E} = {[x] : x \in R}

and

[x] = {y \in R : a n n (x) = a n n (y)}

is called a compressed zero-divisor graph, denoted by

Γ_{E} (R)

. An edge is formed between two vertices

[x]

and

[y]

of

Z (R_{E})

if and only if

[x] [y] = [x y] = [0],

that is, iff

x y = 0

. For a ring

R

, graph

G

is said to be realizable as

Γ_{E} (R)

if

G

is isomorphic to

Γ_{E} (R)

. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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13 pages, 297 KB

Open AccessArticle

On Centralizers of Idempotents with Restricted Range

by Dilawar J. Mir and Amal S. Alali

Symmetry 2024, 16(6), 769; https://doi.org/10.3390/sym16060769 - 19 Jun 2024

Cited by 2 | Viewed by 1397

Abstract

This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of [...] Read more.

This study delves into the structure and properties of left inverse zero divisor bands within semigroups, identifying their maximal forms and broadening the theoretical landscape of semigroup analysis. A significant focus is placed on the automorphisms of a semigroup S of centralizers of idempotent transformations with restricted range, revealing that these automorphisms are inner ones and induced by the units of S. Additionally, we establish that the automorphism group

Aut (S)

is isomorphic to

U_{S}

, the group of units of S. These findings extend previous results on semigroups of transformations, enhancing their applicability and providing a more unified theory. The practical implications of this work span multiple fields, including automata theory, coding theory, cryptography, and graph theory, offering tools for more efficient algorithms and models. By simplifying complex concepts and providing a solid foundation for future research, this study makes significant contributions to both theoretical and applied mathematics. Full article

(This article belongs to the Special Issue Algebraic Systems, Models and Applications)

15 pages, 675 KB

Open AccessArticle

Classification of Genus Three Zero-Divisor Graphs

by Thangaraj Asir, Karuppiah Mano and Turki Alsuraiheed

Symmetry 2023, 15(12), 2167; https://doi.org/10.3390/sym15122167 - 6 Dec 2023

Viewed by 2032

Abstract

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by

Γ (R)

, [...] Read more.

In this paper, we consider the problem of classifying commutative rings according to the genus number of its associating zero-divisor graphs. The zero-divisor graph of R, where R is a commutative ring with nonzero identity, denoted by

Γ (R)

, is the undirected graph whose vertices are the nonzero zero-divisors of R, and the distinct vertices x and y are adjacent if and only if

x y = 0

. Here, we classify the local rings with genus three zero-divisor graphs. Full article

(This article belongs to the Special Issue Advances in Graph Theory)

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17 pages, 311 KB

Open AccessArticle

Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring

Z_{p} \times Z_{t} \times Z_{s}

by Nadeem ur Rehman, Amal S. Alali, Shabir Ahmad Mir and Mohd Nazim

Axioms 2023, 12(10), 987; https://doi.org/10.3390/axioms12100987 - 18 Oct 2023

Cited by 1 | Viewed by 2768

Abstract

Let R be a commutative ring with identity, and

Z (R)

be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by

W Γ (R)

is an undirected (simple) graph with vertex set [...] Read more.

Let R be a commutative ring with identity, and

Z (R)

be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by

W Γ (R)

is an undirected (simple) graph with vertex set

Z {(R)}^{*}

, and two distinct vertices x and y are adjacent, if and only if there exist

r \in a n n (x)

and

s \in a n n (y)

, such that

r s = 0

. Importantly, it is worth noting that

W Γ (R)

contains the zero-divisor graph

Γ (R)

as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring

Z_{p} \times Z_{t} \times Z_{s}

, denoted by

W Γ (Z_{p} \times Z_{t} \times Z_{s})

, where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of

W Γ (Z_{p} \times Z_{t} \times Z_{s})

. Full article

(This article belongs to the Special Issue Recent Advances in Graph Theory with Applications)

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14 pages, 402 KB

Open AccessArticle

On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ_n

by Nazim, Nadeem Ur Rehman and Ahmad Alghamdi

Mathematics 2023, 11(20), 4310; https://doi.org/10.3390/math11204310 - 16 Oct 2023

Cited by 9 | Viewed by 2114

Abstract

For a finite commutative ring

R

with identity

1 \neq 0

, the weakly zero-divisor graph of

R

denoted as

W Γ (R)

is a simple undirected graph having vertex set as a set of non-zero zero-divisors of

R

and two [...] Read more.

For a finite commutative ring

R

with identity

1 \neq 0

, the weakly zero-divisor graph of

R

denoted as

W Γ (R)

is a simple undirected graph having vertex set as a set of non-zero zero-divisors of

R

and two distinct vertices

a

and

b

are adjacent if and only if there exist elements

r \in ann (a)

and

s \in ann (b)

satisfying the condition

r s = 0

. The zero-divisor graph of a ring is a spanning sub-graph of the weakly zero-divisor graph. This article finds the normalized Laplacian spectra of the weakly zero-divisor graph

W Γ (R)

. Specifically, the investigation is carried out on the weakly zero-divisor graph

W Γ (Z_{n})

for various values of

n

. Full article

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25 pages, 1062 KB

Open AccessArticle

On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs

by Amal S. Alali, Shahbaz Ali, Muhammad Adnan and Delfim F. M. Torres

Symmetry 2023, 15(10), 1911; https://doi.org/10.3390/sym15101911 - 12 Oct 2023

Cited by 2 | Viewed by 1797

Abstract

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications [...] Read more.

The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph’s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph

G (Z_{n})

is called a zero-divisor graph over the zero divisors of a commutative ring

Z_{n}

, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded. Full article

(This article belongs to the Special Issue Symmetry in Differential Geometry and Geometric Analysis)

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