ODAM – Vol 6 – Issue 3 (2023) – PISRT https://old.pisrt.org Thu, 18 Jan 2024 11:02:40 +0000 en-US hourly 1 https://wordpress.org/?v=6.7 Chromatically unique \(6\)-bridge graph \(\theta (r,r,s,s,t,u)\) https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/chromatically-unique-6-bridge-graph-theta-rrsstu/ Thu, 18 Jan 2024 11:01:40 +0000 https://old.pisrt.org/?p=8271
ODAM-Vol. 6 (2023), Issue 3, pp. 41 - 56 Open Access Full-Text PDF
Syed Ahtsham Ul Haq Bokhary and Shehr Bano
Abstract: Let \(A\) and \(B\) be two graph and \(P(A,z)\) and \(P(B,z)\) are their chromatic polynomial, respectively. The two graphs \(A\) and \(B\) are said to be chromatic equivalent denoted by \( A \sim B \) if \(P(A,z)=P(B,z)\). A graph \(A\) is said to be chromatically unique(or simply \(\chi\)- unique) if for any graph \(B\) such that \(A\sim B \), we have \(A\cong B\), that is \(A\) is isomorphic to \(B\). In this paper, the chromatic uniqueness of a new family of \(6\)-bridge graph \(\theta(r,r,s,s,t,u)\) where \(2\leq r\leq s \leq t\leq u\) is investigated.]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 41 – 56
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0093

Chromatically unique \(6\)-bridge graph \(\theta (r,r,s,s,t,u)\)

Syed Ahtsham Ul Haq Bokhary\(^{1,*}\), Shehr Bano
\(^{1}\)Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.

Copyright © 2023 Syed Ahtsham Ul Haq Bokhary and Shehr Bano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: July 18, 2023 – Accepted: December 25, 2023 – Published: December 27, 2023

Abstract

Let \(A\) and \(B\) be two graph and \(P(A,z)\) and \(P(B,z)\) are their chromatic polynomial, respectively. The two graphs \(A\) and \(B\) are said to be chromatic equivalent denoted by \( A \sim B \) if \(P(A,z)=P(B,z)\). A graph \(A\) is said to be chromatically unique(or simply \(\chi\)- unique) if for any graph \(B\) such that \(A\sim B \), we have \(A\cong B\), that is \(A\) is isomorphic to \(B\). In this paper, the chromatic uniqueness of a new family of \(6\)-bridge graph \(\theta(r,r,s,s,t,u)\) where \(2\leq r\leq s \leq t\leq u\) is investigated.

Keywords:

Chromatic polynomial; Chromatically unique; multi-bridge
graph.
]]> Connectivity indices and QSPR analysis of benzenoid hydrocarbons https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/connectivity-indices-and-qspr-analysis-of-benzenoid-hydrocarbons/ Wed, 27 Dec 2023 13:55:24 +0000 https://old.pisrt.org/?p=8219
ODAM-Vol. 6 (2023), Issue 3, pp. 35 - 40 Open Access Full-Text PDF
Zhen Lin
Abstract: In mathematical chemistry, a large number of topological indices are used to predict the physicochemical properties of compounds, especially in the study of quantitative structure-proerty relationship (QSPR). However, many topological indices have almost the same predictive ability. In this paper, we focus on how to use fewer topological indices to predict the physicochemical properties of compounds through the QSPR analysis of connectivity indices of benzene hydrocarbons. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 35 – 40
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0092

Connectivity indices and QSPR analysis of benzenoid hydrocarbons

Zhen Lin\(^{1,*}\)
\(^{1}\)School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, Qinghai, China

Copyright © 2023 Zhen Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: November 02, 2023 – Accepted: December 23, 2023 – Published: December 27, 2023

Abstract

In mathematical chemistry, a large number of topological indices are used to predict the physicochemical properties of compounds, especially in the study of quantitative structure-proerty relationship (QSPR).
However, many topological indices have almost the same predictive ability. In this paper, we focus on how to use fewer topological indices to predict the physicochemical properties of compounds through the QSPR analysis of connectivity indices of benzene hydrocarbons.

Keywords:

Connectivity indices; QSPR; Benzene hydrocarbons.
]]> The bounds for topological invariants of a weighted graph using traces https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/the-bounds-for-topological-invariants-of-a-weighted-graph-using-traces/ Tue, 26 Dec 2023 02:32:53 +0000 https://old.pisrt.org/?p=8188
ODAM-Vol. 6 (2023), Issue 3, pp. 30 – 34 Open Access Full-Text PDF
Emre Sevgi, Gül Özkan Kizilirmak and Serife Büyükköse
Abstract: In this paper, we obtain the bounds for the Laplacian eigenvalues of a weighted graph using traces. Then, we find the bounds for the Kirchhoff and Laplacian Estrada indices of a weighted graph. Finally, we define the Laplacian energy of a weighted graph and get the upper bound for this energy. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 30 – 34
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0091

The bounds for topological invariants of a weighted graph using traces

Emre Sevgi\(^{1,*}\), Gül Özkan Kizilirmak\(^{2}\), Serife Büyükköse
\(^{1}\)Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey.; emresevgi@gazi.edu.tr; gulozkangazi.edu.tr; sbuyukkose@gazi.edu.tr

Copyright © 2023 Emre Sevgi, Gül Özkan Kizilirmak and Serife Büyükköse. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: September 15, 2023 – Accepted: November 7, 2023 – Published: December 26, 2023

Abstract

In this paper, we obtain the bounds for the Laplacian eigenvalues of a weighted graph using traces. Then, we find the bounds for the Kirchhoff and Laplacian Estrada indices of a weighted graph. Finally, we define the Laplacian energy of a weighted graph and get the upper bound for this energy.

Keywords:

Laplacian energy; Kirchhoff index; Laplacian Estrada index; Weighted graph.
]]> A note on the characterization of claw-free and paw-free graphs https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/a-note-on-the-characterization-of-claw-free-and-paw-free-graphs/ Tue, 26 Dec 2023 02:28:37 +0000 https://old.pisrt.org/?p=8184
ODAM-Vol. 6 (2023), Issue 3, pp. 26 – 29 Open Access Full-Text PDF
Phillip Mafuta and Josiah Mushanyu
Abstract: A number of results on claw-free, paw-free graphs have been presented in the literature. Although the proofs of such results are elegant, sound and valid, it has gone unnoticed that all the results about claw-free, paw-free graphs in the literature are a consequence of a result by Olariu [1]. The note, apart from covering the aforementioned gap, also provides an alternate proof to a result by Faudree and Gould found in [2] in that, an unnoticed consequence resulted in the characterization of claw-free, paw-free graphs. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 26 – 29
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0090

A note on the characterization of claw-free and paw-free graphs

Phillip Mafuta\(^{1,3,*}\), Josiah Mushanyu\(^{2,3}\)
\(^{1}\)Department of Mathematics and Applied Mathematics IB74, University of the Free State, Bloemfontein, South Africa; phillipmafuta@gmail.com
\(^{2}\)Department of Computing, Mathematical and Statistical Sciences, University of Namibia, Windhoek, Namibia; mushanyuj@gmail.com
\(^{3}\) Department of Mathematics and Computational Sciences, University of Zimbabwe, Harare, Zimbabwe

Copyright © 2023 Rikio Ichishima and Francesc-Antoni Muntaner-Batle. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: November 12, 2023 – Accepted: December 14, 2023 – Published: December 26, 2023

Abstract

A number of results on claw-free, paw-free graphs have been presented in the literature. Although the proofs of such results are elegant, sound and valid, it has gone unnoticed that all the results about claw-free, paw-free graphs in the literature are a consequence of a result by Olariu [1]. The note, apart from covering the aforementioned gap, also provides an alternate proof to a result by Faudree and Gould found in [2] in that, an unnoticed consequence resulted in the characterization of claw-free, paw-free graphs.

Keywords:

Forbidden Sub-graphs; Pancyclicity; Hamiltonicity; Traceability.
]]> On the super edge-magicness of graphs with a specific degree sequence https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/on-the-super-edge-magicness-of-graphs-with-a-specific-degree-sequence/ Tue, 26 Dec 2023 02:26:00 +0000 https://old.pisrt.org/?p=8182
ODAM-Vol. 6 (2023), Issue 3, pp. 22 – 25 Open Access Full-Text PDF
Rikio Ichishima and Francesc-Antoni Muntaner-Batle
Abstract: A graph \(G\) is said to be super edge-magic if there exists a bijective function \(f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}\) such that \(f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}\) and \(f\left(u\right) + f\left(v\right) + f\left(uv\right)\) is a constant for each \(uv\in E\left( G\right)\). In this paper, we study the super edge-magicness of graphs of order \(n\) with degree sequence \(s:4, 2, 2, \ldots, 2\). We also investigate the super edge-magic properties of certain families of graphs. This leads us to propose some open problems. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 22 – 25
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0089

On the super edge-magicness of graphs with a specific degree sequence

Rikio Ichishima\(^{1,*}\), Francesc-Antoni Muntaner-Batle\(^{2}\)
\(^{1}\)Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, Japan.; ichishim@kokushikan.ac.jp
\(^{2}\)Graph Theory and Applications Research Group, School of Electrical Engineering and Computer Science, Faculty of Engineering and Built Environment, The University of Newcastle, NSW 2308 Australia.; famb1es@yahoo.es

Copyright © 2023 Rikio Ichishima and Francesc-Antoni Muntaner-Batle. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: August 14, 2023 – Accepted: September 26, 2023 – Published: December 26, 2023

Abstract

A graph \(G\) is said to be super edge-magic if there exists a bijective function \(f:V\left(G\right) \cup E\left(G\right)\rightarrow \left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert +\left\vert E\left( G\right) \right\vert \right\}\) such that \(f\left(V \left(G\right)\right) =\left\{1, 2, \ldots , \left\vert V\left( G\right) \right\vert \right\}\) and \(f\left(u\right) + f\left(v\right) + f\left(uv\right)\) is a constant for each \(uv\in E\left( G\right)\). In this paper, we study the super edge-magicness of graphs of order \(n\) with degree sequence \(s:4, 2, 2, \ldots, 2\). We also investigate the super edge-magic properties of certain families of graphs. This leads us to propose some open problems.

Keywords:

(Super) edge-magic graph; (Super) edge-magic labeling; Vertex degree; Degree sequence; Graph labeling.
]]> Lattice path coronoids https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-3-2023/lattice-path-coronoids/ Tue, 26 Dec 2023 02:20:28 +0000 https://old.pisrt.org/?p=8180
ODAM-Vol. 6 (2023), Issue 3, pp. 1 – 21 Open Access Full-Text PDF
Tricia Muldoon Brown
Abstract: Coronoids are nice chemical structures that may be represented mathematically in the planar hexagonal lattice. They have been well-studied both for their chemical properties and also their enumerative aspects. Typical approaches to the latter type of questions often include classification and algorithmic techniques. Here we study one simple class of coronoids called hollow hexagons. Notably, hollow hexagons may be represented with a collection of partitions on the set \(\{2,3,4,6\}\). The hollow hexagons are used to classify another family of primitive coronoids, which we introduce here, called lattice path coronoids. Techniques from lattice path enumeration are used to count these newly-defined structures within equivalence classes indexed by enclosing hollow hexagons.]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 3, pp. 1 – 21
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0088

Lattice path coronoids

Tricia Muldoon Brown\(^{1,*}\)
\(^{1}\)Department of Mathematical Sciences, Georgia Southern University, Savannah, GA, U.S.A.; tmbrown@georgiasouthern.edu

Copyright © 2023 Tricia Muldoon Brown. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Received: July 6, 2023 – Accepted: December 13, 2023 – Published: December 26, 2023

Abstract

Coronoids are nice chemical structures that may be represented mathematically in the planar hexagonal lattice. They have been well-studied both for their chemical properties and also their enumerative aspects. Typical approaches to the latter type of questions often include classification and algorithmic techniques. Here we study one simple class of coronoids called hollow hexagons. Notably, hollow hexagons may be represented with a collection of partitions on the set \(\{2,3,4,6\}\). The hollow hexagons are used to classify another family of primitive coronoids, which we introduce here, called lattice path coronoids. Techniques from lattice path enumeration are used to count these newly-defined structures within equivalence classes indexed by enclosing hollow hexagons.

Keywords:

Primitive coronoid; Lattice path; Partition.
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