Mathematics

Research

13 pages, 276 KiB

Open AccessArticle

An Inverse Approach for Finding Graphs with a Failed Zero Forcing Number of k

by Chirag Kaudan, Rachel Taylor and Darren A. Narayan

Mathematics 2023, 11(19), 4068; https://doi.org/10.3390/math11194068 - 25 Sep 2023

Viewed by 1063

Abstract

For a given a graph G, the zero forcing number of G,

Z (G)

, is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being included [...] Read more.

For a given a graph G, the zero forcing number of G,

Z (G)

, is the smallest cardinality of any set S of vertices on which repeated applications of the forcing rule results in all vertices being included in S. The forcing rule is as follows: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. The failed zero forcing number of a graph was introduced by Fetcie, Jacob, and Saavedra and was defined as the cardinality of the largest set of vertices which fails to force all vertices in the graph. In 2021, Gomez et al. proved that there were exactly 15 graphs with a failed zero forcing number of two, but their proof was complicated, requiring the analysis of many graph families. We present an “inverse” approach where we start with a set of vertices S and then see which graphs have S as a maximum-sized failed zero forcing set. This results in not only a shorter proof but characterizes which graphs have a particular failed zero forcing set. In our proof, we characterize the graphs which have a failed zero forcing set S where

| S | = 2

, meaning considering all simple graphs of order two as induced subgraphs. This approach also has greater potential for characterizing graphs where

F (G) > 2

since many general arguments on the structure of graphs are developed in this type of analysis and are applicable for failed zero forcing sets of any size. Full article

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

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22 pages, 1176 KiB

Open AccessArticle

A Conceptual Graph-Based Method to Compute Information Content

by Rolando Quintero, Miguel Torres-Ruiz, Magdalena Saldaña-Pérez, Carlos Guzmán Sánchez-Mejorada and Felix Mata-Rivera

Mathematics 2023, 11(18), 3972; https://doi.org/10.3390/math11183972 - 19 Sep 2023

Cited by 1 | Viewed by 1021

Abstract

This research uses the computing of conceptual distance to measure information content in Wikipedia categories. The proposed metric, generality, relates information content to conceptual distance by determining the ratio of the information that a concept provides to others compared to the information that [...] Read more.

This research uses the computing of conceptual distance to measure information content in Wikipedia categories. The proposed metric, generality, relates information content to conceptual distance by determining the ratio of the information that a concept provides to others compared to the information that it receives. The DIS-C algorithm calculates generality values for each concept, considering each relationship’s conceptual distance and distance weight. The findings of this study are compared to current methods in the field and found to be comparable to results obtained using the WordNet corpus. This method offers a new approach to measuring information content applied to any relationship or topology in conceptualization. Full article

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

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10 pages, 256 KiB

Open AccessArticle

Enhanced Effectiveness in Various Ladder Graphs Based on the F-Centroidal Meanness Criterion

by A. Rajesh Kannan, S. Murali Krishnan, Karuppusamy Loganathan, Nazek Alessa and M. Hymavathi

Mathematics 2023, 11(14), 3205; https://doi.org/10.3390/math11143205 - 21 Jul 2023

Cited by 3 | Viewed by 1087

Abstract

Graph labeling allows for the representation of additional attributes or properties associated with the vertices, edges, or both of graphs. This can provide a more comprehensive and detailed representation of the system being modeled, allowing for a richer analysis and interpretation of the [...] Read more.

Graph labeling allows for the representation of additional attributes or properties associated with the vertices, edges, or both of graphs. This can provide a more comprehensive and detailed representation of the system being modeled, allowing for a richer analysis and interpretation of the graph. Graph labeling in ladder graphs has a wide range of applications in engineering, computer science, physics, biology, and other fields. It can be applied to various problem domains, such as image processing, wireless sensor networks, VLSI design, bioinformatics, social network analysis, transportation networks, and many others. The versatility of ladder graphs and the ability to apply graph labeling to them make them a powerful tool for modeling and analyzing diverse systems. If a function

Υ

is an injective vertex assignment in

{1, 2, \dots q + 1}

and the inductive edge assignment function

Υ^{*}

in

{1, 2, \dots q}

is expressed as a graph with q edges, defined as

Υ^{*} (u v) = ⌊\frac{2 [Υ {(u)}^{2} + Υ (u) Υ (v) + Υ {(v)}^{2}]}{3 [Υ (u) + Υ (v)]}⌋,

then the function is referred to as F-centroidal mean labeling. This is known as the F-centroidal mean criterion. Here, we have determined the F-centroidal mean criteria of the graph ladder, slanting ladder, triangular ladder,

T L_{n} \circ S_{m}

,

S L_{n} \circ S_{m}

for

m \leq 2

, double-sided step ladder,

D_{n}^{*}

, and diamond ladder. Full article

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

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15 pages, 578 KiB

Open AccessArticle

On Characterization of Balance and Consistency Preserving d-Antipodal Signed Graphs

by Kshittiz Chettri and Biswajit Deb

Mathematics 2023, 11(13), 2982; https://doi.org/10.3390/math11132982 - 4 Jul 2023

Viewed by 933

Abstract

A signed graph is an ordered pair

Σ = (G, σ)

, where G is a graph and

σ : E (G) ⟶ {+ 1, - 1}

is a mapping. For [...] Read more.

A signed graph is an ordered pair

Σ = (G, σ)

, where G is a graph and

σ : E (G) ⟶ {+ 1, - 1}

is a mapping. For

e \in E (G)

,

σ (e)

is called the sign of e and for any sub-graph H of G,

σ (H) = \prod_{e \in E (H)} σ (e)

is called the sign of H. A signed graph having a sign of each cycle

+ 1

is called balanced. Two vertices in a graph G are called antipodal if

d_{G} (u, v) = d i a m (G)

. The antipodal graph

A (G)

of a graph G is the graph with a vertex set that is the same as that of G, and two vertices

u, v

in

A (G)

are adjacent if

u, v

are antipodal. By the d-antipodal graph

G_{d}^{A}

of a graph G, we refer to the union of G and

A (G)

. Given a signed graph

Σ = (G, σ)

, the signed graph

Σ_{d}^{A} = (G_{d}^{A}, σ_{d})

is called the d-antipodal signed graph of G, where

σ_{d}

is defined as follows:

σ_{d} (e) = σ (e) if e \in E (G) and otherwise, σ_{d} (e) = \prod_{P \in P_{e}} σ (P)

, where

P_{e}

is the collection of all diametric paths in

Σ

connecting the end vertices of an antipodal edge e in

Σ_{d}^{A}

. In this article, the balance property and canonical consistency of d-antipodal signed graphs of Smith signed graphs (connected graphs having a highest eigenvalue of 2) are studied. Full article

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

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12 pages, 374 KiB

Open AccessArticle

The Crossing Number of Join of a Special Disconnected 6-Vertex Graph with Cycle

by Zongpeng Ding and Xiaomei Qian

Mathematics 2023, 11(10), 2253; https://doi.org/10.3390/math11102253 - 11 May 2023

Cited by 1 | Viewed by 1192

Abstract

The crossing number of a graph G,

c r (G)

, is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. There are almost no results concerning crossing number of join of a [...] Read more.

The crossing number of a graph G,

c r (G)

, is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. There are almost no results concerning crossing number of join of a disconnected 6-vertex graph with cycle. The main aim of this paper is to give the crossing number of the join product

Q + C_{n}

for the disconnected 6-vertex graph Q consisting of the two 3-cycles, where

C_{n}

is the cycle on n vertices. Full article

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

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14 pages, 456 KiB

Open AccessArticle

Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling

by Sakander Hayat and Farwa Asmat

Mathematics 2023, 11(10), 2245; https://doi.org/10.3390/math11102245 - 11 May 2023

Cited by 15 | Viewed by 1609

Abstract

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph

Ω

provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first [...] Read more.

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph

Ω

provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph

Ω

is defined as the product of the sum of the

α

th powers of the vertex degrees of

Ω

, where

α

is a real number such that

α \neq 0

and

α \neq 1

. The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index

Π_{1}^{α}

for predicting the

π

-electronic energy

E_{π} (β)

of benzenoid hydrocarbons. In particular, we answer the question concerning the value of

α

for which the predictive potential of

Π_{1}^{α}

with

E_{π}

for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that

Π_{1}^{α}

correlates with

E_{π}

of lower benzenoid hydrocarbons with correlation coefficient

ρ = - 0.998

, if

α = - 0.00496

. In QSPR modeling, the value

ρ = - 0.998

is considered to be considerably significant. Full article

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

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22 pages, 409 KiB

Open AccessArticle

Structures of Critical Nontree Graphs with Cutwidth Four

by Zhenkun Zhang and Hongjian Lai

Mathematics 2023, 11(7), 1631; https://doi.org/10.3390/math11071631 - 28 Mar 2023

Viewed by 1592

Abstract

The cutwidth of a graph G is the smallest integer k (

k \geq 1

) such that the vertices of G are arranged in a linear layout

[v_{1}, v_{2}, . . ., v_{n}]

, [...] Read more.

The cutwidth of a graph G is the smallest integer k (

k \geq 1

) such that the vertices of G are arranged in a linear layout

[v_{1}, v_{2}, . . ., v_{n}]

, in such a way that for each

i = 1, 2, . . ., n - 1

, there are at most k edges with one endpoint in

{v_{1}, v_{2}, . . ., v_{i}}

and the other in

{v_{i + 1}, . . ., v_{n}}

. The cutwidth problem for G is to determine the cutwidth k of G. A graph G with cutwidth k is k-cutwidth critical if every proper subgraph of G has a cutwidth less than k and G is homeomorphically minimal. In this paper, except five irregular graphs, other 4-cutwidth critical graphs were resonably classified into two classes, which are graph class with a central vertex

v_{0}

, and graph class with a central cycle

C_{q}

of length

q \leq 6

, respectively, and any member of two graph classes can skillfuly achieve a subgraph decomposition

S

with cardinality 2, 3 or 4, where each member of

S

is either a 2-cutwith graph or a 3-cutwidth graph. Full article

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

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9 pages, 300 KiB

Open AccessArticle

Fixing Numbers of Point-Block Incidence Graphs

by Josephine Brooks, Alvaro Carbonero, Joseph Vargas, Rigoberto Flórez, Brendan Rooney and Darren A. Narayan

Mathematics 2023, 11(6), 1322; https://doi.org/10.3390/math11061322 - 9 Mar 2023

Viewed by 1250

Abstract

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in [...] Read more.

A vertex in a graph is referred to as fixed if it is mapped to itself under every automorphism of the vertices. The fixing number of a graph is the minimum number of vertices, when fixed, that fixes all of the vertices in the graph. Fixing numbers were first introduced by Laison and Gibbons, and independently by Erwin and Harary. Fixing numbers have also been referred to as determining numbers by Boutin. The main motivation is to remove all symmetries from a graph. A very simple application is in the creation of QR codes where the symbols must be fixed against any rotation. We determine the fixing number for several families of graphs, including those arising from combinatorial block designs. We also present several infinite families of graphs with an even stronger condition, where fixing any vertex in a graph fixes every vertex. Full article

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

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12 pages, 2219 KiB

Open AccessArticle

Authentication of Counterfeit Hundred Ringgit Malaysian Banknotes Using Fuzzy Graph Method

by Nurfarhana Hassan, Tahir Ahmad, Naji Arafat Mahat, Hasmerya Maarof, Mujahid Abdullahi, Nur Farah Dina Ajid, Zarith Sofia Jasmi and Foo Keat How

Mathematics 2023, 11(4), 1002; https://doi.org/10.3390/math11041002 - 16 Feb 2023

Viewed by 2315

Abstract

A banknote is a currency issued by a country, and it was first introduced in the 16th century. The counterfeiting of banknotes by cunning criminals became a great challenge with the current advanced technology. Forensic scientists are using chemical methods, such as Fourier [...] Read more.

A banknote is a currency issued by a country, and it was first introduced in the 16th century. The counterfeiting of banknotes by cunning criminals became a great challenge with the current advanced technology. Forensic scientists are using chemical methods, such as Fourier transform infrared (FTIR) spectroscopy for differentiating genuine and counterfeit banknotes. However, the FTIR spectra of banknotes may require further pattern recognition analysis due to their high similarities. In this paper, a fuzzy graph-based algorithm for authentication of the FTIR spectrum, namely chemometrics fuzzy autocatalytic set (c-FACS), is used to discriminate between genuine and counterfeit hundred Ringgit Malaysian (RM100) banknotes. The results show that the genuine and counterfeit RM100 banknotes have slightly distinct patterns when analyzed using c-FACS. In addition, the results are compared with RM50 banknotes, and the results reveal that the nodes or dominant axis varies between the two banknotes. To verify the reliability of the results, the results obtained via c-FACS are compared with principal component analysis (PCA). The c-FACS showed better performances as compared to PCA in terms of time consumption and observation. Thus, the c-FACS has the ability to assist forensic investigations involving banknote counterfeiting crimes. Full article

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

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11 pages, 839 KiB

Open AccessArticle

Generalized Quasi Trees with Respect to Degree Based Topological Indices and Their Applications to COVID-19 Drugs

by Alaa Altassan and Muhammad Imran

Mathematics 2023, 11(3), 647; https://doi.org/10.3390/math11030647 - 27 Jan 2023

Cited by 2 | Viewed by 1371

Abstract

The l-generalized quasi tree is a graph G for which we can find

W \subset V (G)

with

| W | = l

such that

G - W

is a tree but for an arbitrary [...] Read more.

The l-generalized quasi tree is a graph G for which we can find

W \subset V (G)

with

| W | = l

such that

G - W

is a tree but for an arbitrary

Y \subset V (G)

with

| Y | < l

,

G - Y

is not a tree. In this paper, inequalities with respect to zeroth-order Randić and hyper-Zagreb indices are studied in the class of l-generalized quasi trees. The corresponding extremal graphs corresponding to these indices in the class of l-generalized quasi trees are also obtained. In addition, we carry QSPR analysis of COVID-19 drugs with zeroth-order Randić and hyper-Zagreb indices (energy). Full article

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

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13 pages, 290 KiB

Open AccessArticle

On Bond Incident Degree Indices of Chemical Graphs

by Abeer M. Albalahi, Akbar Ali, Zhibin Du, Akhlaq Ahmad Bhatti, Tariq Alraqad, Naveed Iqbal and Amjad E. Hamza

Mathematics 2023, 11(1), 27; https://doi.org/10.3390/math11010027 - 21 Dec 2022

Cited by 6 | Viewed by 1567

Abstract

By swapping out atoms for vertices and bonds for edges, a graph may be used to model any molecular structure. A graph G is considered to be a chemical graph in graph theory if no vertex of G has a degree of 5 [...] Read more.

By swapping out atoms for vertices and bonds for edges, a graph may be used to model any molecular structure. A graph G is considered to be a chemical graph in graph theory if no vertex of G has a degree of 5 or greater. The bond incident degree (BID) index for a chemical graph G is defined as the total of contributions

f (d_{G} (u), d_{G} (v))

from all edges

u v

of G, where

d_{G} (w)

stands for the degree of a vertex w of G,

E (G)

is the set of edges of G, and f is a real-valued symmetric function. This paper addresses the problem of finding graphs with extremum BID indices over the class of all chemical graphs of a fixed number of edges and vertices. Full article

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

13 pages, 2559 KiB

Open AccessArticle

Inverse Sum Indeg Index (Energy) with Applications to Anticancer Drugs

by Alaa Altassan, Bilal Ahmad Rather and Muhammad Imran

Mathematics 2022, 10(24), 4749; https://doi.org/10.3390/math10244749 - 14 Dec 2022

Cited by 7 | Viewed by 2211

Abstract

For a simple graph with vertex set

{v_{1}, v_{2}, \dots, v_{n}}

with degree sequence

d_{v_{i}}

of vertex

v_{i}, i = 1, 2, \dots, n

, the inverse sum [...] Read more.

For a simple graph with vertex set

{v_{1}, v_{2}, \dots, v_{n}}

with degree sequence

d_{v_{i}}

of vertex

v_{i}, i = 1, 2, \dots, n

, the inverse sum indeg matrix (

I S I

-matrix)

A_{I S I} (G) = {(a_{i j})}_{n \times n}

of G is defined by

a_{i j} = \frac{d_{v_{i}} d_{v_{j}}}{d_{v_{i}} + d_{v_{j}}},

if

v_{i}

is adjacent to

v_{j}

, and zero, otherwise. The multiset of eigenvalues of

A_{I S I} (G)

is the

I S I

-spectrum of G and the sum of their absolute values is the

I S I

-energy of

G

. In this paper, we modify the two results of (Li, Ye and Broersma, 2022), give the correct characterization of the extremal graphs and thereby obtain better bounds than the already known results. Moreover, we also discuss the QSPR analysis and carry the statistical modelling (linear, logarithmic and quadratic) of the physicochemical properties of anticancer drugs with the

I S I

-index (energy). Full article

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

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9 pages, 490 KiB

Open AccessArticle

Characterization of All Graphs with a Failed Skew Zero Forcing Number of 1

by Aidan Johnson, Andrew E. Vick and Darren A. Narayan

Mathematics 2022, 10(23), 4463; https://doi.org/10.3390/math10234463 - 26 Nov 2022

Viewed by 1324

Abstract

Given a graph G, the zero forcing number of G,

Z (G)

, is the minimum cardinality of any set S of vertices of which repeated applications of the forcing rule results in all vertices being in S. [...] Read more.

Given a graph G, the zero forcing number of G,

Z (G)

, is the minimum cardinality of any set S of vertices of which repeated applications of the forcing rule results in all vertices being in S. The forcing rule is: if a vertex v is in S, and exactly one neighbor u of v is not in S, then u is added to S in the next iteration. Hence the failed zero forcing number of a graph was defined to be the cardinality of the largest set of vertices which fails to force all vertices in the graph. A similar property called skew zero forcing was defined so that if there is exactly one neighbor u of v is not in S, then u is added to S in the next iteration. The difference is that vertices that are not in S can force other vertices. This leads to the failed skew zero forcing number of a graph, which is denoted by

F^{-} (G)

. In this paper, we provide a complete characterization of all graphs with

F^{-} (G) = 1

. Fetcie, Jacob, and Saavedra showed that the only graphs with a failed zero forcing number of 1 are either: the union of two isolated vertices;

P_{3}

;

K_{3}

; or

K_{4}

. In this paper, we provide a surprising result: changing the forcing rule to a skew-forcing rule results in an infinite number of graphs with

F^{-} (G) = 1

. Full article

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

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12 pages, 568 KiB

Open AccessArticle

Computing Sharp Bounds of Metric Based Fractional Dimensions for the Sierpinski Networks

by Arooba Fatima, Ahmed Alamer and Muhammad Javaid

Mathematics 2022, 10(22), 4332; https://doi.org/10.3390/math10224332 - 18 Nov 2022

Cited by 1 | Viewed by 1771

Abstract

The concept of metric dimension is widely applied to solve various problems in the different fields of computer science and chemistry, such as computer networking, integer programming, robot navigation, and the formation of chemical structuring. In this article, the local fractional metric dimension [...] Read more.

The concept of metric dimension is widely applied to solve various problems in the different fields of computer science and chemistry, such as computer networking, integer programming, robot navigation, and the formation of chemical structuring. In this article, the local fractional metric dimension (LFMD) of the cycle-based Sierpinski networks is computed with the help of its local resolving neighborhoods of all the adjacent pairs of vertices. In addition, the boundedness of LFMD is also examined as the order of the Sierpinski networks approaches infinity. Full article

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

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18 pages, 583 KiB

Open AccessFeature PaperArticle

Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model

by Liudmila Zhilyakova, Vasily Koreshkov and Nadezhda Chaplinskaia

Mathematics 2022, 10(21), 4095; https://doi.org/10.3390/math10214095 - 3 Nov 2022

Cited by 2 | Viewed by 2314

Abstract

The resource network is a non-linear threshold model where vertices exchange resource in infinite discrete time. The model is represented by a directed weighted graph. At each time step, all vertices send their resources along all output edges following one of two rules. [...] Read more.

The resource network is a non-linear threshold model where vertices exchange resource in infinite discrete time. The model is represented by a directed weighted graph. At each time step, all vertices send their resources along all output edges following one of two rules. For each vertex, the threshold value for changing the operation rule is equal to the total weight of its outgoing edges. If all vertices have resources less than their thresholds, the network is completely described by a homogeneous Markov chain. If at least one of the vertices has a resource above the threshold, the network is described by a non-homogeneous Markov chain. The purpose of this article is to describe and investigate non-homogeneous Markov chains generated by the resource network model. It is proven that they are strongly ergodic. In addition, stochastic matrices of a special form were studied. A number of new properties were revealed for them. The results obtained were generalized to arbitrary stochastic matrices. Full article

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

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13 pages, 288 KiB

Open AccessArticle

The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products

by Alaa Altassan, Muhammad Imran and Shehnaz Akhter

Mathematics 2022, 10(16), 2834; https://doi.org/10.3390/math10162834 - 9 Aug 2022

Viewed by 2904

Abstract

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric [...] Read more.

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products. Full article

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

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16 pages, 785 KiB

Open AccessArticle

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

by Sakander Hayat, Asad Khan and Yubin Zhong

Mathematics 2022, 10(11), 1815; https://doi.org/10.3390/math10111815 - 25 May 2022

Cited by 12 | Viewed by 2062

Abstract

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, which provides an infinite [...] Read more.

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, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs. Full article

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

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18 pages, 971 KiB

Open AccessArticle

The IRC Indices of Transformation and Derived Graphs

by Haichang Luo, Sakander Hayat, Yubin Zhong, Zhongyuan Peng and Tamás Réti

Mathematics 2022, 10(7), 1111; https://doi.org/10.3390/math10071111 - 30 Mar 2022

Cited by 3 | Viewed by 2072

Abstract

An irregularity index

I R (Γ)

of a graph

Γ

is a nonnegative numeric quantity (i.e.,

I R (Γ) \geq 0

) such that

I R (Γ) = 0

iff

Γ

is a regular graph. In this [...] Read more.

An irregularity index

I R (Γ)

of a graph

Γ

is a nonnegative numeric quantity (i.e.,

I R (Γ) \geq 0

) such that

I R (Γ) = 0

iff

Γ

is a regular graph. In this paper, we show that

I R C

closely correlates with the normal boiling point

T_{b p}

and the standard heat of formation

Δ H_{f}^{o}

of lower benzenoid hydrocarbons. The correlation models that fit the data efficiently for both

T_{b p}

and

Δ H_{f}^{o}

are linear. We develop further mathematical properties of

I R C

by calculating its exact expressions for the recently introduced transformation graphs as well as certain derived graphs, such as the total graph, semi-total point graph, subdivision graph, semi-total line graph, double, strong double, and extended double cover graphs. Some open problems are proposed for further research on the

I R C

index of graphs. Full article

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

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