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16 pages, 487 KiB

Open AccessArticle

The General Extended Adjacency Eigenvalues of Chain Graphs

by Bilal Ahmad Rather, Hilal A. Ganie, Kinkar Chandra Das and Yilun Shang

Mathematics 2024, 12(2), 192; https://doi.org/10.3390/math12020192 - 6 Jan 2024

Cited by 3 | Viewed by 1105

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds [...] Read more.

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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25 pages, 3510 KiB

Open AccessArticle

Algebraic Structure Graphs over the Commutative Ring

Z_{m}

: Exploring Topological Indices and Entropies Using

M

-Polynomials

by Amal S. Alali, Shahbaz Ali, Noor Hassan, Ali M. Mahnashi, Yilun Shang and Abdullah Assiry

Mathematics 2023, 11(18), 3833; https://doi.org/10.3390/math11183833 - 7 Sep 2023

Cited by 17 | Viewed by 1932

Abstract

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures [...] Read more.

The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph

\hat{G}

is fully made up of the zero divisors of the modular ring

Z_{n}

, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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15 pages, 454 KiB

Open AccessArticle

General Atom-Bond Sum-Connectivity Index of Graphs

by Abeer M. Albalahi, Emina Milovanović and Akbar Ali

Mathematics 2023, 11(11), 2494; https://doi.org/10.3390/math11112494 - 29 May 2023

Cited by 9 | Viewed by 1980

Abstract

This paper is concerned with the general atom-bond sum-connectivity index

A B S_{γ}

, which is a generalization of the recently proposed atom-bond sum-connectivity index, where

γ

is any real number. For a connected graph G with more than two vertices, the [...] Read more.

This paper is concerned with the general atom-bond sum-connectivity index

A B S_{γ}

, which is a generalization of the recently proposed atom-bond sum-connectivity index, where

γ

is any real number. For a connected graph G with more than two vertices, the number

A B S_{γ} (G)

is defined as the sum of

{(1 - 2 {(d_{x} + d_{y})}^{- 1})}^{γ}

over all edges

x y

of the graph G, where

d_{x}

and

d_{y}

represent the degrees of the vertices x and y of G, respectively. For

- 10 \leq γ \leq 10

, the significance of

A B S_{γ}

is examined on the data set of twenty-five benzenoid hydrocarbons for predicting their enthalpy of formation. It is found that the predictive ability of the index

A B S_{γ}

for the selected property of the considered hydrocarbons is comparable to other existing general indices of this type. The effect of the addition of an edge between two non-adjacent vertices of a graph under

A B S_{γ}

is also investigated. Furthermore, several extremal results regarding trees, general graphs, and triangle-free graphs of a given number of vertices are proved. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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12 pages, 671 KiB

Open AccessArticle

A Distributed Algorithm for the Assignment of the Laplacian Spectrum for Path Graphs

by Gianfranco Parlangeli

Mathematics 2023, 11(10), 2359; https://doi.org/10.3390/math11102359 - 18 May 2023

Cited by 3 | Viewed by 1246

Abstract

In this paper, we bring forward a distributed algorithm for the assignment of a prescribed Laplacian spectrum for path graphs by means of asymmetric weight assignment. We first describe the meaningfulness and the relevance of this mathematical setting in modern technological applications, and [...] Read more.

In this paper, we bring forward a distributed algorithm for the assignment of a prescribed Laplacian spectrum for path graphs by means of asymmetric weight assignment. We first describe the meaningfulness and the relevance of this mathematical setting in modern technological applications, and some examples are reported, revealing its practical usefulness in a variety of applications. Then, the solution is derived both theoretically and through an algorithm. Special attention is devoted to a distributed implementation of the main algorithm, which is a valuable feature for several modern applications. Finally, the positivity is discussed, which is revealed to be a consequence of the strict interlacing property. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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16 pages, 360 KiB

Open AccessArticle

Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

by Laxman Saha, Bapan Das, Kalishankar Tiwary, Kinkar Chandra Das and Yilun Shang

Mathematics 2023, 11(8), 1896; https://doi.org/10.3390/math11081896 - 17 Apr 2023

Cited by 6 | Viewed by 1183

Abstract

Let

G = (V (G), E (G))

be a simple connected unweighted graph. A set

R \subset V (G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of [...] Read more.

Let

G = (V (G), E (G))

be a simple connected unweighted graph. A set

R \subset V (G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set

S_{x, y} = {w \in R : d (w, x) \neq d (w, y)}

is at least k for every pair of distinct vertices

x, y

of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph

C (n : 1, 2)

for all possible values of k and

n .

The optimal fault-tolerant resolving sets with k tolerance are also delineated. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

18 pages, 455 KiB

Open AccessArticle

On a Combinatorial Approach to Studying the Steiner Diameter of a Graph and Its Line Graph

by Hongfang Liu, Zhizhang Shen, Chenxu Yang and Kinkar Chandra Das

Mathematics 2022, 10(20), 3863; https://doi.org/10.3390/math10203863 - 18 Oct 2022

Cited by 2 | Viewed by 1418

Abstract

In 1989, Chartrand, Oellermann, Tian and Zou introduced the Steiner distance for graphs. This is a natural generalization of the classical graph distance concept. Let

Γ

be a connected graph of order at least 2, and

S \ V (Γ)

. [...] Read more.

In 1989, Chartrand, Oellermann, Tian and Zou introduced the Steiner distance for graphs. This is a natural generalization of the classical graph distance concept. Let

Γ

be a connected graph of order at least 2, and

S \ V (Γ)

. Then, the minimum size among all the connected subgraphs whose vertex sets contain S is the Steiner distance

d_{Γ} (S)

among the vertices of S. The Steiner k-eccentricity $e_{k} (v)$ of a vertex v of

Γ

is defined by

e_{k} (v) = max {d_{Γ} (S) | S \ V (Γ), | S | = k, a n d v \in S}

, where n and k are two integers, with

2 \leq k \leq n

, and the Steiner k-diameter of

Γ

is defined by

{sdiam}_{k} (Γ) = max {e_{k} (v) | v \in V (Γ)}

. In this paper, we present an algorithm to derive the Steiner distance of a graph; in addition, we obtain a relationship between the Steiner k-diameter of a graph and its line graph. We study various properties of the Steiner diameter through a combinatorial approach. Moreover, we characterize graph

Γ

when

{sdiam}_{k} (Γ)

is given, and we determine

{sdiam}_{k} (Γ)

for some special graphs. We also discuss some of the applications of Steiner diameter, including one in education networks. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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18 pages, 343 KiB

Open AccessArticle

On General Reduced Second Zagreb Index of Graphs

by Lkhagva Buyantogtokh, Batmend Horoldagva and Kinkar Chandra Das

Mathematics 2022, 10(19), 3553; https://doi.org/10.3390/math10193553 - 29 Sep 2022

Cited by 4 | Viewed by 1731

Abstract

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant

G R M_{α}

, known under the name general reduced second [...] Read more.

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant

G R M_{α}

, known under the name general reduced second Zagreb index, is defined as

G R M_{α} (Γ) = \sum_{u v \in E (Γ)} (d_{Γ} (u) + α) (d_{Γ} (v) + α)

, where

d_{Γ} (v)

is the degree of the vertex v of the graph

Γ

and

α

is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to

G R M_{α}

(α \geq - \frac{1}{2})

. Using the extremal unicyclic graphs, we obtain a lower bound on

G R M_{α} (Γ)

of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on

G R M_{α}

of different classes of graphs in terms of order n, size m, independence number

γ

, chromatic number k, etc. In particular, we present an upper bound on

G R M_{α}

of connected triangle-free graph of order

n > 2

,

m > 0

edges with

α > - 1.5

, and characterize the extremal graphs. Finally, we prove that the Turán graph

T_{n} (k)

gives the maximum

G R M_{α} (α \geq - 1)

among all graphs of order n with chromatic number k. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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10 pages, 331 KiB

Open AccessArticle

Extra Edge Connectivity and Extremal Problems in Education Networks

by Hongfang Liu, Jinxia Liang and Kinkar Chandra Das

Mathematics 2022, 10(19), 3475; https://doi.org/10.3390/math10193475 - 23 Sep 2022

Cited by 1 | Viewed by 1480

Abstract

Extra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The p-extra edge connectivity

λ_{p} (Γ)

of a graph

Γ

was introduced by Fàbrega and Fiol in 1996. In this paper, we [...] Read more.

Extra edge connectivity and diagnosability have been employed to investigate the fault tolerance properties of network structures. The p-extra edge connectivity

λ_{p} (Γ)

of a graph

Γ

was introduced by Fàbrega and Fiol in 1996. In this paper, we find the exact values of p-extra edge connectivity of some special graphs. Moreover, we give some upper and lower bounds for

λ_{p} (Γ)

, and graphs with

λ_{p} (Γ) = 1, 2, ⌈\frac{n}{2}⌉ ⌊\frac{n}{2}⌋ - 1, ⌈\frac{n}{2}⌉ ⌊\frac{n}{2}⌋

are characterized. Finally, we obtain the three extremal results for the p-extra edge connectivity. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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