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Open AccessArticle

Maximum General Sum-Connectivity Index of Trees and Unicyclic Graphs with Given Order and Number of Pendant Vertices

by Elize Swartz and Tomáš Vetrík

Mathematics 2025, 13(19), 3061; https://doi.org/10.3390/math13193061 - 23 Sep 2025

Viewed by 307

For

a \in R

, the general sum-connectivity index of a graph G is defined as [...] Read more.

For

a \in R

, the general sum-connectivity index of a graph G is defined as

χ_{a} (G) = \sum_{u v \in E (G)} {[d_{G} (u) + d_{G} (v)]}^{a}

, where

E (G)

is the set of edges of G and

d_{G} (u)

and

d_{G} (v)

are the degrees of vertices u and v, respectively. For trees and unicyclic graphs with given order and number of pendant vertices, we present upper bounds on the general sum-connectivity index

χ_{a}

, where

0 < a < 1

. We also present the trees and unicyclic graphs that attain the maximum general sum-connectivity index for

0 < a < 1

. Full article

(This article belongs to the Special Issue Topological Analysis and Computation of Chemical Graphs and Physical Networks)

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9 pages, 1015 KB

Open AccessArticle

Extremal Values of Second Zagreb Index of Unicyclic Graphs Having Maximum Cycle Length: Two New Algorithms

by Hacer Ozden Ayna

Mathematics 2025, 13(15), 2475; https://doi.org/10.3390/math13152475 - 31 Jul 2025

Viewed by 420

Abstract

It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could be stated as the condition that [...] Read more.

It is well-known that the necessary and sufficient condition for a connected graph to be unicyclic is that its omega invariant, a recently introduced graph invariant useful in combinatorial and topological calculations, is zero. This condition could be stated as the condition that the order and the size of the graph are equal. Using a recent result saying that the length of the unique cycle could be any integer between 1 and n − a₁ where a₁ is the number of pendant vertices in the graph, two explicit labeling algorithms are provided that attain these extremal values of the first and second Zagreb indices by means of an application of the well-known rearrangement inequality. When the cycle has the maximum length, we obtain the situation where all the pendant vertices are adjacent to the support vertices, the neighbors of the pendant vertices, which are placed only on the unique cycle. This makes it easy to calculate the second Zagreb index, as the contribution of the pendant edges to such indices is fixed, implying that we can only calculate these indices for the edges on the cycle. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 3rd Edition)

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15 pages, 302 KB

Open AccessArticle

Extremal Permanents of Laplacian Matrices of Unicyclic Graphs

by Tingzeng Wu, Xiuhong Wang and Xiangshuai Dong

Axioms 2025, 14(8), 565; https://doi.org/10.3390/axioms14080565 - 24 Jul 2025

Viewed by 288

Abstract

The extremal problem of Laplacian permanents of graphs is a classical and challenging topic in algebraic combinatorics, where the inherent #P-complete complexity of permanent computation renders this pursuit particularly intractable. In this paper, we determine the upper and lower bounds of permanents of [...] Read more.

The extremal problem of Laplacian permanents of graphs is a classical and challenging topic in algebraic combinatorics, where the inherent #P-complete complexity of permanent computation renders this pursuit particularly intractable. In this paper, we determine the upper and lower bounds of permanents of Laplacian matrices of unicyclic graphs, and the corresponding extremal graphs are characterized. Furthermore, we also determine the upper and lower bounds of permanents of Laplacian matrices of unicyclic graphs with given girth, and the corresponding extremal graphs are characterized. Full article

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33 pages, 3435 KB

Open AccessArticle

Investigation of General Sombor Index for Optimal Values in Bicyclic Graphs, Trees, and Unicyclic Graphs Using Well-Known Transformations

by Miraj Khan, Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Symmetry 2025, 17(6), 968; https://doi.org/10.3390/sym17060968 - 18 Jun 2025

Viewed by 854

Abstract

The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the idea of optimization in [...] Read more.

The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the idea of optimization in the families of bicyclic graphs, trees, and unicyclic graphs. We investigated optimal values in the stated families by means of well-known transformations. The transformations include the following: Transformation A, Transformation B, Transformation C, and Transformation D. Transformation A and Transformation B increase the value of the generalized Sombor index, while Transformation C and Transformation D are used for minimal values. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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11 pages, 239 KB

Open AccessEditor’s ChoiceArticle

Resolving an Open Problem on the Exponential Arithmetic–Geometric Index of Unicyclic Graphs

by Kinkar Chandra Das and Jayanta Bera

Mathematics 2025, 13(9), 1391; https://doi.org/10.3390/math13091391 - 24 Apr 2025

Cited by 1 | Viewed by 427

Abstract

Recently, the exponential arithmetic–geometric index (

E A G

) was introduced. The exponential arithmetic–geometric index (

E A G

) of a graph G is defined as [...] Read more.

Recently, the exponential arithmetic–geometric index (

E A G

) was introduced. The exponential arithmetic–geometric index (

E A G

) of a graph G is defined as

E A G (G) = \sum_{v_{i} v_{j} \in E (G)} e^{\frac{d_{i} + d_{j}}{2 \sqrt{d_{i} d_{j}}}}

, where

d_{i}

represents the degree of the vertex

v_{i}

in G. The characterization of extreme structures in relation to graph invariants from the class of unicyclic graphs is an important problem in discrete mathematics. Cruz et al., 2022 proposed a unified method for finding extremal unicyclic graphs for exponential degree-based graph invariants. However, in the case of

E A G

, this method is insufficient for generating the maximal unicyclic graph. Consequently, the same article presented an open problem for the investigation of the maximal unicyclic graph with respect to this invariant. This article completely characterizes the maximal unicyclic graph in relation to

E A G

. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

12 pages, 369 KB

Open AccessArticle

Extremal Unicyclic Graphs for the Euler Sombor Index: Applications to Benzenoid Hydrocarbons and Drug Molecules

by Zhenhua Su and Zikai Tang

Axioms 2025, 14(4), 249; https://doi.org/10.3390/axioms14040249 - 26 Mar 2025

Cited by 1 | Viewed by 625

Abstract

With geometric significance, the Euler Sombor index of a graph

Γ

is defined as [...] Read more.

With geometric significance, the Euler Sombor index of a graph

Γ

is defined as

E P (Γ) = \sum_{{u v} \in E (Γ)} \sqrt{d {(u)}^{2} + d {(v)}^{2} + d (u) d (v)}

. It originates from the mathematical distance property and has been proven to have good chemical applications in octane isomers. In this paper, the minimum and maximum of the Euler Sombor index for unicyclic graphs with given girth, as well as the corresponding extremal graphs, are determined. As an application, the experimental values of this index for some benzenoid hydrocarbons and drug molecules were compared with the boiling point. Through regression analysis, it was further demonstrated that the Euler Sombor index has excellent predictability in the physicochemical properties of compounds. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization and Discrete Mathematics with Applications)

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28 pages, 897 KB

Open AccessFeature PaperArticle

Metric Locations in Pseudotrees: A Survey and New Results

by José Cáceres and Ignacio M. Pelayo

Mathematics 2025, 13(4), 560; https://doi.org/10.3390/math13040560 - 8 Feb 2025

Viewed by 667

Abstract

This paper presents a comprehensive review of the literature on the original concept of metric location, along with its various adaptations and extensions that have been developed over time. Given that determining a minimum location set is generally NP-hard, we focus on analyzing [...] Read more.

This paper presents a comprehensive review of the literature on the original concept of metric location, along with its various adaptations and extensions that have been developed over time. Given that determining a minimum location set is generally NP-hard, we focus on analyzing the behavior of these sets within specific graph families, including paths, cycles, trees and unicyclic graphs. In addition to synthesizing existing knowledge, we contribute new findings and insights to the field, advancing the understanding of metric location problems in these structured graph classes. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

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19 pages, 308 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Exponential Atom-Bond Connectivity Index of Graphs

by Kinkar Chandra Das

Mathematics 2025, 13(2), 269; https://doi.org/10.3390/math13020269 - 15 Jan 2025

Cited by 4 | Viewed by 2265

Abstract

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological [...] Read more.

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows:

e^{ABC} = e^{ABC} (Υ) = \sum_{v_{i} v_{j} \in E (Υ)} e^{\sqrt{\frac{d_{i} + d_{j} - 2}{d_{i} d_{j}}}},

where

d_{i}

is the degree of the vertex

v_{i}

in

Υ

. In this paper, we prove that the double star

D S_{n - 3, 1}

is the second maximal graph with respect to the

e^{ABC}

index of trees of order n. We give an upper bound on

e^{ABC}

of unicyclic graphs of order n and characterize the maximal graphs. The graph

K_{1} \lor (P_{3} \cup (n - 4) K_{1})

gives the maximal graph with respect to the

e^{ABC}

index of bicyclic graphs of order n. We present several relations between

e^{ABC} (Υ)

and

A B C (Υ)

of graph

Υ

. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research. Full article

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35 pages, 3375 KB

Open AccessArticle

Optimization in Symmetric Trees, Unicyclic Graphs, and Bicyclic Graphs with Help of Mappings Using Second Form of Generalized Power-Sum Connectivity Index

by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Symmetry 2025, 17(1), 122; https://doi.org/10.3390/sym17010122 - 15 Jan 2025

Cited by 1 | Viewed by 1018

Abstract

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. [...] Read more.

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is

Ω

Y_{α} (Ω) = \sum_{ζ ϱ \in E (Ω)} {(d_{Ω} {(ζ)}^{d_{Ω} (ζ)} + d_{Ω} {(ϱ)}^{d_{Ω} (ϱ)})}^{α}

, while the second form of the GPSCI is defined as

Y_{β} (Ω) = \sum_{ζ ϱ \in E (Ω)} {(d_{Ω} {(ζ)}^{d_{Ω} (ζ)} \times d_{Ω} {(ϱ)}^{d_{Ω} (ϱ)})}^{β}

. In this paper, we investigate

Y_{β}

in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for

Y_{β}

using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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33 pages, 3247 KB

Open AccessArticle

Optimization of General Power-Sum Connectivity Index in Uni-Cyclic Graphs, Bi-Cyclic Graphs and Trees by Means of Operations

by Muhammad Yasin Khan, Gohar Ali and Ioan-Lucian Popa

Axioms 2024, 13(12), 840; https://doi.org/10.3390/axioms13120840 - 28 Nov 2024

Cited by 1 | Viewed by 993

Abstract

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for [...] Read more.

The field of indices has been explored and advanced by various researchers for different purposes. One purpose is the optimization of indices in various problems. In this work, the general power-sum connectivity index is considered. The general power-sum connectivity index was investigated for k-generalized quasi-trees where optimal graphs were found. Further, in this work, we extend the idea of optimization to families of graphs, including uni-cyclic graphs, bi-cyclic graphs and trees. The optimization is carried out by means of operations named as Operation A, B, C and D. The first two operations increase the value of the general power-sum connectivity index, while the last two work opposite to Operations A and B. These operations are explained by means of diagrams, where one can easily obtain their working procedures. Full article

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

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12 pages, 376 KB

Open AccessArticle

On Unicyclic Graphs with a Given Number of Pendent Vertices or Matching Number and Their Graphical Edge-Weight-Function Indices

by Akbar Ali, Abdulaziz M. Alanazi, Taher S. Hassan and Yilun Shang

Mathematics 2024, 12(23), 3658; https://doi.org/10.3390/math12233658 - 22 Nov 2024

Cited by 2 | Viewed by 1381

Abstract

Consider a unicyclic graph G with edge set

E (G)

. Let

f

be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G [...] Read more.

Consider a unicyclic graph G with edge set

E (G)

. Let

f

be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as

I_{f} (G) = \sum_{x y \in E (G)} f (d_{G} (x), d_{G} (y))

, where

d_{G} (x)

denotes the degree a vertex x in G. This paper determines optimal bounds for

I_{f} (G)

in terms of the order of G and a parameter

z

, where

z

is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function

f

must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices. Full article

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24 pages, 2445 KB

Open AccessArticle

On Some Distance Spectral Characteristics of Trees

by Sakander Hayat, Asad Khan and Mohammed J. F. Alenazi

Axioms 2024, 13(8), 494; https://doi.org/10.3390/axioms13080494 - 23 Jul 2024

Cited by 2 | Viewed by 1165

Abstract

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study [...] Read more.

Graham and Pollack in 1971 presented applications of eigenvalues of the distance matrix in addressing problems in data communication systems. Spectral graph theory employs tools from linear algebra to retrieve the properties of a graph from the spectrum of graph-theoretic matrices. The study of graphs with “few eigenvalues” is a contemporary problem in spectral graph theory. This paper studies graphs with few distinct distance eigenvalues. After mentioning the classification of graphs with one and two distinct distance eigenvalues, we mainly focus on graphs with three distinct distance eigenvalues. Characterizing graphs with three distinct distance eigenvalues is “highly” non-trivial. In this paper, we classify all trees whose distance matrix has precisely three distinct eigenvalues. Our proof is different from earlier existing proof of the result as our proof is extendable to other similar families such as unicyclic and bicyclic graphs. The main tools which we employ include interlacing and equitable partitions. We also list all the connected graphs on ν ≤ 6 vertices and compute their distance spectra. Importantly, all these graphs on ν ≤ 6 vertices are determined from their distance spectra. We deliver a distance cospectral pair of order 7, thus making it a distance cospectral pair of the smallest order. This paper is concluded with some future directions. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)

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17 pages, 2999 KB

Open AccessFeature PaperArticle

On Unicyclic Graphs with Minimum Graovac–Ghorbani Index

by Snježana Majstorović Ergotić

Mathematics 2024, 12(3), 384; https://doi.org/10.3390/math12030384 - 24 Jan 2024

Cited by 2 | Viewed by 2141

Abstract

In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of molecular problems. Its main goal is to [...] Read more.

In discrete mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Chemical graph theory is concerned with non-trivial applications of graph theory to the solution of molecular problems. Its main goal is to use numerical invariants to reduce the topological structure of a molecule to a single number that characterizes its properties. Topological indices are numerical invariants associated with the chemical constitution, for the purpose of the correlation of chemical structures with various physical properties, chemical reactivity, or biological activity. They have found important application in predicting the behavior of chemical substances. The Graovac–Ghorbani (

A B C_{G G}

) index is a topological descriptor that has improved predictive potential compared to analogous descriptors. It is used to model both the boiling point and melting point of molecules and is applied in the pharmaceutical industry. In the recent years, the number of publications on its mathematical properties has increased. The aim of this work is to partially solve an open problem, namely to find the structure of unicyclic graphs that minimize the

A B C_{G G}

index. We characterize unicyclic graphs with even girth that minimize the

A B C_{G G}

index, while we also present partial results for odd girths. As an auxiliary result, we compare the

A B C_{G G}

indices of paths and cycles with an odd number of vertices. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory, 3rd Edition)

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12 pages, 412 KB

Open AccessArticle

General Randić Index of Unicyclic Graphs and Its Applications to Drugs

by Alaa Altassan and Muhammad Imran

Symmetry 2024, 16(1), 113; https://doi.org/10.3390/sym16010113 - 18 Jan 2024

Cited by 3 | Viewed by 2098

Abstract

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs with a fixed maximum degree

Δ

and the maximum and the second maximum [...] Read more.

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs with a fixed maximum degree

Δ

and the maximum and the second maximum general Randić index for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs, where

η_{0} \approx - 0.21

. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes. Full article

(This article belongs to the Section Mathematics)

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14 pages, 497 KB

Open AccessArticle

A Unified Approach for Extremal General Exponential Multiplicative Zagreb Indices

by Rashad Ismail, Muhammad Azeem, Yilun Shang, Muhammad Imran and Ali Ahmad

Axioms 2023, 12(7), 675; https://doi.org/10.3390/axioms12070675 - 9 Jul 2023

Cited by 17 | Viewed by 2266

Abstract

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are [...] Read more.

The study of the maximum and minimal characteristics of graphs is the focus of the significant field of mathematics known as extreme graph theory. Finding the biggest or smallest graphs that meet specified criteria is the main goal of this discipline. There are several applications of extremal graph theory in various fields, including computer science, physics, and chemistry. Some of the important applications include: Computer networking, social networking, chemistry and physics as well. Recently, in 2021 exponential multiplicative Zagreb indices were introduced. In generalization, we introduce the generalized form of exponential multiplicative Zagreb indices for

α \in R^{+} \ {1} .

Furthermore, to see the behaviour of generalized first and second exponential Zagreb indices for

α \in R^{+} \ {1},

we used a transformation method. In term of the two newly developed generalized exponential multiplicative Zagreb indices, we will investigate the extremal bicyclic, uni-cyclic and trees graphs. Four graph transformations are used and some bounds are presented in terms of generalized exponential multiplicative Zagreb indices. Full article

(This article belongs to the Special Issue Spectral Graph Theory, Molecular Graph Theory and Their Applications)

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