ODAM – Vol 6 – Issue 2 (2023) – PISRT https://old.pisrt.org Thu, 26 Oct 2023 16:42:03 +0000 en-US hourly 1 https://wordpress.org/?v=6.7 On 2-noncrossing increasing trees https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/on-2-noncrossing-increasing-trees/ Mon, 23 Oct 2023 18:40:29 +0000 https://old.pisrt.org/?p=8030
ODAM-Vol. 6 (2023), Issue 2, pp. 39 – 50 Open Access Full-Text PDF
Isaac Owino Okoth
Abstract:A \(2\)-noncrossing tree is a rooted tree drawn in the plane with its vertices (colored black or white) on the boundary of a circle such that the edges are line segments that do not intersect inside the circle and there is no black-black ascent in any path from the root. A rooted tree is said to be increasing if the labels of the vertices are increasing as one moves away from the root. In this paper, we use generating functions and bijections to enumerate \(2\)-noncrossing increasing trees by the number of blacks vertices and by root degree. Bijections with noncrossing trees, ternary trees, 2-plane trees, certain Dyck paths, and certain restricted lattice paths are established.]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 2, pp. 39 – 50
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0087

On 2-noncrossing increasing trees

Isaac Owino Okoth
Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya.; ookoth@maseno.ac.ke

Copyright © 2023 Isaac Owino Okoth. 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: August 30, 2023 – Published: October 23, 2023

Abstract

A \(2\)-noncrossing tree is a rooted tree drawn in the plane with its vertices (colored black or white) on the boundary of a circle such that the edges are line segments that do not intersect inside the circle and there is no black-black ascent in any path from the root. A rooted tree is said to be increasing if the labels of the vertices are increasing as one moves away from the root. In this paper, we use generating functions and bijections to enumerate \(2\)-noncrossing increasing trees by the number of blacks vertices and by root degree. Bijections with noncrossing trees, ternary trees, 2-plane trees, certain Dyck paths, and certain restricted lattice paths are established.

Keywords:

2-noncrossing tree; increasing tree; 2-plane tree; Dyck path; lattice path
]]> Introduction to total chromatic vertex stress of graphs https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/introduction-to-total-chromatic-vertex-stress-of-graphs/ Mon, 23 Oct 2023 18:27:38 +0000 https://old.pisrt.org/?p=8028
ODAM-Vol. 6 (2023), Issue 2, pp. 32 – 38 Open Access Full-Text PDF
Johan Kok
Abstract:This paper introduces the new notion of total chromatic vertex stress of a graph. Results for certain tree families and other \(2\)-colorable graphs are presented. The notions of chromatically-stress stability and chromatically-stress regularity are also introduced. New research avenues are also proposed.]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 2, pp. 32 – 38
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0086

Introduction to total chromatic vertex stress of graphs

Johan Kok
Independent Mathematics Researcher, City of Tshwane, South Africa &
Visiting Faculty at CHRIST (Deemed to be a University), Bangalore, India.

Copyright © 2023 Johan Kok. 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: March 13, 2023 – Accepted: July 05, 2023 – Published: October 23, 2023

Abstract

This paper introduces the new notion of total chromatic vertex stress of a graph. Results for certain tree families and other \(2\)-colorable graphs are presented. The notions of chromatically-stress stability and chromatically-stress regularity are also introduced. New research avenues are also proposed.

Keywords:

Chromatic vertex stress; chromatic equivalent coloring; chromatically-stress stable; chromatically-stress regular
]]> Arc coloring of odd graphs for hamiltonicity https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/arc-coloring-of-odd-graphs-for-hamiltonicity/ Mon, 23 Oct 2023 18:08:02 +0000 https://old.pisrt.org/?p=8025
ODAM-Vol. 6 (2023), Issue 2, pp. 14 – 31 Open Access Full-Text PDF
Italo Dejter
Abstract:Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for \(1< k \in\mathbb{Z}\), we find that the odd graph \(O_k\) has an arc factorization with colors \(0,1,\ldots,k\) such that the sum of colors of the two arcs of each edge equals \(k\). This is applied to analyzing the influence of such arc factorizations in recently constructed uniform 2-factors in \(O_k\) and in Hamilton cycles in \(O_k\) as well as in its double covering graph known as the middle-levels graph \(M_k\). ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 2, pp. 14 – 31
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0085

Arc coloring of odd graphs for hamiltonicity

Italo Dejter
Department of Mathematics, University of Puerto Rico, San Juan, Puerto Rico.; italo.dejter@gmail.com

Copyright © 2023 Italo Dejter. 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: May 8, 2023 – Accepted: June 16, 2023 – Published: October 23, 2023

Abstract

Coloring the arcs of biregular graphs was introduced with possible applications to industrial chemistry, molecular biology, cellular neuroscience, etc. Here, we deal with arc coloring in some non-bipartite graphs. In fact, for \(1<k\in\mathbb{Z}\), we find that the odd graph \(O_k\) has an arc factorization with colors \(0,1,\ldots,k\) such that the sum of colors of the two arcs of each edge equals \(k\). This is applied to analyzing the influence of such arc factorizations in recently constructed uniform 2-factors in \(O_k\) and in Hamilton cycles in \(O_k\) as well as in its double covering graph known as the middle-levels graph \(M_k\).

Keywords:

Arc coloring; Hamilton cycle; odd graphs; k-germs; Dyck words
]]> More on second Zagreb energy of graphs https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/more-on-second-zagreb-energy-of-graphs/ Mon, 23 Oct 2023 17:51:54 +0000 https://old.pisrt.org/?p=8021
ODAM-Vol. 6 (2023), Issue 2, pp. 7 – 13 Open Access Full-Text PDF
Mitesh J. Patel, Kajal S. Baldaniya and Ashika Panicker
Abstract:Let \(G\) be a graph with \(n\) vertices. The second Zagreb energy of graph \(G\) is defined as the sum of the absolute values of the eigenvalues of the second Zagreb matrix of graph \(G\). In this paper, we derive the relation between the second Zagreb matrix and the adjacency matrix of graph \(G\) and derive the new upper bound for the second Zagreb energy in the context of trace. We also derive the second Zagreb energy of \(m-\)splitting graph and \(m-\)shadow graph of a graph. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 2, pp. 7 – 13
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0084

More on second Zagreb energy of graphs

Mitesh J. Patel\(^{1,*}\), Kajal S. Baldaniya\(^{2}\) and Ashika Panicker\(^{1}\)
\(^{1}\) Department of Mathematics, Tolani College of Arts and Science, Adipur- Kachchh, Gujarat – INDIA.; miteshmaths1984@gmail.com (M.J.P) ; panickerashika@gmail.com (A.P)
\(^{2}\) Department of Mathematics, Gajwani Institute of Science and Technology, Adipur- Kachchh, Gujarat – INDIA.; kajalbaldaniya20@gmail.com (K.S.B)

Copyright © 2023 Mitesh J. Patel, Kajal S. Baldaniya and Ashika Panicker. 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: March 14, 2023 – Accepted: August 05, 2023 – Published: October 23, 2023

Abstract

Let \(G\) be a graph with \(n\) vertices. The second Zagreb energy of graph \(G\) is defined as the sum of the absolute values of the eigenvalues of the second Zagreb matrix of graph \(G\). In this paper, we derive the relation between the second Zagreb matrix and the adjacency matrix of graph \(G\) and derive the new upper bound for the second Zagreb energy in the context of trace. We also derive the second Zagreb energy of \(m-\)splitting graph and \(m-\)shadow graph of a graph.

Keywords:

Second Zagreb energy; \(m-\)splitting graph; \(m-\)shadow graph; regular graph
]]> On the product of Sombor and modified Sombor indices https://old.pisrt.org/psr-press/journals/odam-vol-6-issue-2-2023/on-the-product-of-sombor-and-modified-sombor-indices/ Mon, 23 Oct 2023 17:18:31 +0000 https://old.pisrt.org/?p=8015
ODAM-Vol. 6 (2023), Issue 2, pp. 1 – 6 Open Access Full-Text PDF
Ivan Gutman, Redžepović and Boris Furtula
Abstract:The Sombor index (\(SO\)) and the modified Sombor index (\(^mSO\)) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product \(SO \cdot {^mSO}\) and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either \(SO\) or \(^mSO\). Lower and upper bounds for \(SO \cdot {^mSO}\) are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product \(SO \cdot {^mSO}\) is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point. ]]>
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Open Journal of Discrete Applied Mathematics
Vol. 6 (2023), Issue 2, pp. 1 – 6
ISSN: 2617-9687 (Online) 2617-9679 (Print)
DOI: 10.30538/psrp-odam2023.0083

On the product of Sombor and modified Sombor indices

Ivan Gutman\(^{1,*}\), Izudin Redžepović\(^{2}\) and Boris Furtula\(^{1}\)
\(^{1}\) Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia; gutman@kg.ac.rs; (I. G.) furtula@uni.kg.ac.rs (B. F.)
\(^{2}\) State University of Novi Pazar, 36300 Novi Pazar, Serbia; iredzepovic@np.ac.rs (I.R.)

Copyright © 2023 Ivan Gutman, Redžepović and Boris Furtula . 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: May 22, 2023 – Accepted: August 04, 2023 – Published: October 23, 2023

Abstract

The Sombor index (\(SO\)) and the modified Sombor index (\(^mSO\)) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product \(SO \cdot {^mSO}\) and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either \(SO\) or \(^mSO\). Lower and upper bounds for \(SO \cdot {^mSO}\) are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product \(SO \cdot {^mSO}\) is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point.

Keywords:

Sombor index; modified Sombor index; topological index; degree (of vertex).
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