Symmetry

Research

16 pages, 358 KiB

Open AccessArticle

Study on Structural Properties of Brain Networks Based on Independent Set Indices

by Anagha Puthanpurakkal and Selvakumar Ramachandran

Symmetry 2023, 15(5), 1032; https://doi.org/10.3390/sym15051032 - 6 May 2023

Viewed by 1098

Studies of brain network organisation have swiftly adopted graph theory-based quantitative analysis of complicated networks. Small-world topology, densely connected hubs, and modularity characterise the brain’s structural and functional systems. Many measures quantify graph topology. It has not yet been determined which measurements are [...] Read more.

Studies of brain network organisation have swiftly adopted graph theory-based quantitative analysis of complicated networks. Small-world topology, densely connected hubs, and modularity characterise the brain’s structural and functional systems. Many measures quantify graph topology. It has not yet been determined which measurements are most appropriate for brain network analysis. This work introduces a new parameter applicable to brain network analysis. This parameter may help in the identification of symmetry and the study of symmetry breakdown in the brain. This is important because decreased symmetry in the brain is associated with a decreased chance of developing neurodevelopmental and psychiatric disorders. This work is to study brain networks using maximal independent set-based topological indices. These indices seem to depict significant properties of brain networks, such as clustering, small-worldness, etc. One new parameter introduced in this paper for brain network analysis depends on Zagreb topological indices and independence degree. This parameter is useful for analyzing clusters, rich clubs, small-worldness, and connectivity in modules. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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13 pages, 746 KiB

Open AccessArticle

Valency-Based Indices for Some Succinct Drugs by Using M-Polynomial

by Muhammad Usman Ghani, Francis Joseph H. Campena, K. Pattabiraman, Rashad Ismail, Hanen Karamti and Mohamad Nazri Husin

Symmetry 2023, 15(3), 603; https://doi.org/10.3390/sym15030603 - 27 Feb 2023

Cited by 15 | Viewed by 1672

Abstract

A topological index, which is a number, is connected to a graph. It is often used in chemometrics, biomedicine, and bioinformatics to anticipate various physicochemical properties and biological activities of compounds. The purpose of this article is to encourage original research focused on [...] Read more.

A topological index, which is a number, is connected to a graph. It is often used in chemometrics, biomedicine, and bioinformatics to anticipate various physicochemical properties and biological activities of compounds. The purpose of this article is to encourage original research focused on topological graph indices for the drugs azacitidine, decitabine, and guadecitabine as well as an investigation of the genesis of symmetry in actual networks. Symmetry is a universal phenomenon that applies nature’s conservation rules to complicated systems. Although symmetry is a ubiquitous structural characteristic of complex networks, it has only been seldom examined in real-world networks. The

\bar{M}

-polynomial, one of these polynomials, is used to create a number of degree-based topological coindices. Patients with higher-risk myelodysplastic syndromes, chronic myelomonocytic leukemia, and acute myeloid leukemia who are not candidates for intense regimens, such as induction chemotherapy, are treated with these hypomethylating drugs. Examples of these drugs are decitabine (5-aza-20-deoxycytidine), guadecitabine, and azacitidine. The

\bar{M}

-polynomial is used in this study to construct a variety of coindices for the three brief medicines that are suggested. New cancer therapies could be developed using indice knowledge, specifically the first Zagreb index, second Zagreb index, F-index, reformulated Zagreb index, modified Zagreb, symmetric division index, inverse sum index, harmonic index, and augmented Zagreb index for the drugs azacitidine, decitabine, and guadecitabine. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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17 pages, 1118 KiB

Open AccessArticle

Enumerating Subtrees of Flower and Sunflower Networks

by Long Li, Zongpu Jia, Yu Yang, Fengge Duan, Hailian Lv and Weiting Zhao

Symmetry 2023, 15(2), 284; https://doi.org/10.3390/sym15020284 - 19 Jan 2023

Viewed by 1280

Abstract

Symmetry widely exists in many complex and real-world networks, with flower networks and sunflower networks being two richly symmetric networks and having many practical applications due to their special structures. The number of subtrees (the subtree number index) is closely related to the [...] Read more.

Symmetry widely exists in many complex and real-world networks, with flower networks and sunflower networks being two richly symmetric networks and having many practical applications due to their special structures. The number of subtrees (the subtree number index) is closely related to the reliable network design. Using a generating function, structural analysis techniques, and auxiliary structure introduction, this paper presents the subtree generating functions of flower networks

F l_{n, m} (n \geq 3, m \geq 2)

and sunflower networks

S f_{n, m} (n \geq 3, m \geq 2)

and, thus, solves the computation of subtree number indices of

F l_{n, m} (n \geq 3, m \geq 2)

and

S f_{n, m} (n \geq 3, m \geq 2)

. The results provide a fundamental and efficient method for exploring novel features of symmetric complex cyclic networks from the structural subtree number index perspective. For instance, we conclude that under some parameter constraints, the flower networks are more reliable than sunflower networks. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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21 pages, 372 KiB

Open AccessArticle

The Generalized Inverse Sum Indeg Index of Some Graph Operations

by Ying Wang, Sumaira Hafeez, Shehnaz Akhter, Zahid Iqbal and Adnan Aslam

Symmetry 2022, 14(11), 2349; https://doi.org/10.3390/sym14112349 - 8 Nov 2022

Cited by 3 | Viewed by 1431

Abstract

The study of networks and graphs carried out by topological measures performs a vital role in securing their hidden topologies. This strategy has been extremely used in biomedicine, cheminformatics and bioinformatics, where computations dependent on graph invariants have been made available to communicate [...] Read more.

The study of networks and graphs carried out by topological measures performs a vital role in securing their hidden topologies. This strategy has been extremely used in biomedicine, cheminformatics and bioinformatics, where computations dependent on graph invariants have been made available to communicate the various challenging tasks. In quantitative structure–activity (QSAR) and quantitative structure–property (QSPR) relationship studies, topological invariants are brought into practical action to associate the biological and physicochemical properties and pharmacological activities of materials and chemical compounds. In these studies, the degree-based topological invariants have found a significant position among the other descriptors due to the ease of their computing process and the speed with which these computations can be performed. Thereby, assessing these invariants is one of the flourishing lines of research. The generalized form of the degree-based inverse sum indeg index has recently been introduced. Many degree-based topological invariants can be derived from the generalized form of this index. In this paper, we provided the bounds related to this index for some graph operations, including the Kronecker product, join, corona product, Cartesian product, disjunction, and symmetric difference. We also presented the exact formula of this index for the disjoint union, linking, and splicing of graphs. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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16 pages, 355 KiB

Open AccessArticle

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

by Fawad Ali, Bilal A. Rather, Nahid Fatima, Muhammad Sarfraz, Asad Ullah, Khalid Abdulkhaliq M. Alharbi and Rahim Dad

Symmetry 2022, 14(6), 1266; https://doi.org/10.3390/sym14061266 - 19 Jun 2022

Cited by 13 | Viewed by 1966

Abstract

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It [...] Read more.

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider

G

as a group and H as a non-empty subset of

G

. The commuting graph

C (G, H)

, has H as the vertex set, where

h_{1}, h_{2} \in H

are edge connected whenever

h_{1}

and

h_{2}

commute in

G

. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of

SL (2, C)

. Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

32 pages, 530 KiB

Open AccessArticle

Forbidden Pairs of Disconnected Graphs for Traceability of Block-Chains

by Wanpeng Lei, Liming Xiong, Junfeng Du and Jun Yin

Symmetry 2022, 14(6), 1221; https://doi.org/10.3390/sym14061221 - 13 Jun 2022

Viewed by 1508

Abstract

Each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. It occurs to us that forbidden subgraph conditions for a [...] Read more.

Each traceable graph must be a block-chain; however, a block-chain is not necessarily traceable in general. Whether a given graph is a block-chain or not can be easily verified by a polynomial algorithm. It occurs to us that forbidden subgraph conditions for a block-chain are traceable. In this article, we characterize all pairs of disconnected forbidden subgraphs for the traceability of block-chains, so as to completely solve pairs of forbidden subgraphs for the traceability of block-chains (including disconnected and connected). Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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10 pages, 364 KiB

Open AccessArticle

On Spectral Characterization of Two Classes of Unicycle Graphs

by Jun Yin, Haixing Zhao, Xiujuan Ma and Jing Liang

Symmetry 2022, 14(6), 1213; https://doi.org/10.3390/sym14061213 - 12 Jun 2022

Cited by 1 | Viewed by 1255

Abstract

Let G be a graph with n vertices, let

A (G)

be an adjacency matrix of G and let

P_{A} (G, λ)

be the characteristic polynomial of

A (G)

. The adjacency spectrum of G [...] Read more.

Let G be a graph with n vertices, let

A (G)

be an adjacency matrix of G and let

P_{A} (G, λ)

be the characteristic polynomial of

A (G)

. The adjacency spectrum of G consists of eigenvalues of

A (G)

. A graph G is said to be determined by its adjacency spectrum (DS for short) if other graphs with the same adjacency spectrum as G are isomorphic to G. In this paper, we investigate the spectral characterization of unicycle graphs with only two vertices of degree three. We use

G_{21} (s_{1}, s_{2})

to denote the graph obtained from

Q (s_{1}, s_{2})

by identifying its pendant vertex and the vertex of degree two of

P_{3}

, where

Q (s_{1}, s_{2})

is the graph obtained by identifying a vertex of

C_{s_{1}}

and a pendant vertex of

P_{s_{2}}

. We use

G_{31} (t_{1}, t_{2})

to denote the graph obtained from circle with the vertices

v_{0} v_{1} \dots v_{t_{1} + t_{2} + 1}

by adding one pendant edge at vertices

v_{0}

and

v_{t_{1} + 1}

, respectively. It is shown that

G_{21} (s_{1}, s_{2})

(

s_{1} \neq 4, 6

,

s_{1} \geq 3

,

s_{2} \geq 3

) and

G_{31} (t_{1}, t_{2})

(

t_{1} + t_{2} \neq 2

,

t_{2} \geq t_{1} \geq 1

) are determined by their adjacency spectrum. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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12 pages, 279 KiB

Open AccessArticle

A Complete Characterization of Bidegreed Split Graphs with Four Distinct α-Eigenvalues

by Guifu Su, Guanbang Song, Jun Yin and Junfeng Du

Symmetry 2022, 14(5), 899; https://doi.org/10.3390/sym14050899 - 28 Apr 2022

Cited by 1 | Viewed by 1414

Abstract

It is a well-known fact that a graph of diameter d has at least

d + 1

eigenvalues. A graph is d-extremal (resp.

d_{α}

-extremal) if it has diameter d and exactly

d + 1

distinct eigenvalues (resp.

α

-eigenvalues), and [...] Read more.

It is a well-known fact that a graph of diameter d has at least

d + 1

eigenvalues. A graph is d-extremal (resp.

d_{α}

-extremal) if it has diameter d and exactly

d + 1

distinct eigenvalues (resp.

α

-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either

\tilde{d}

or

\hat{d}

, then we say it is

(\tilde{d}, \hat{d})

-bidegreed. In this paper, we present a complete classification of the connected bidegreed

3_{α}

-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

8 pages, 294 KiB

Open AccessArticle

Super Vertex (Edge)-Connectivity of Varietal Hypercube

by Zhecheng Yu, Liqiong Xu, Shanshan Yin and Litao Guo

Symmetry 2022, 14(2), 304; https://doi.org/10.3390/sym14020304 - 2 Feb 2022

Cited by 6 | Viewed by 1482

Abstract

The reliability measure of networks is of significant importance to the design and maintenance of networks. Based on connectivity, many refined quantitative indicators for the reliability of network systems have been introduced. The super vertex edge-connectivity and cyclic edge-connectivity, as important parameters to [...] Read more.

The reliability measure of networks is of significant importance to the design and maintenance of networks. Based on connectivity, many refined quantitative indicators for the reliability of network systems have been introduced. The super vertex edge-connectivity and cyclic edge-connectivity, as important parameters to evaluate the robustness of networks, are explored extensively. As a variant of the hypercube

Q_{n}

, the varietal hypercube

V Q_{n}

has better properties than

Q_{n}

with the same number of edges and vertices. Wang and Xu have proved that

V Q_{n}

is super vertex-connected for

n \geq 1

and is also super edge-connected if

n \neq 2

. In this paper, we use another method to prove these results. Moreover, we also obtain the super restricted connectivity and the cyclic edge-connectivity of the varietal hypercube

V Q_{n}

. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

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