Volume 2023 Issue 3

Author(s): Syed Ahtsham Ul Haq Bokhary1, Shehr Bano1
1Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan.
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.

Author(s): Zhen Lin1
1School of Mathematics and Statistics, Qinghai Normal University, Xining, 810008, Qinghai, China
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.

Author(s): Emre Sevgi1, Gül Özkan Kizilirmak, Serife Büyükköse
1Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey.
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.

Author(s): Phillip Mafuta1,2, Josiah Mushanyu3,2
1Department of Mathematics and Applied Mathematics IB74, University of the Free State, Bloemfontein, South Africa
2Department of Mathematics and Computational Sciences, University of Zimbabwe, Harare, Zimbabwe
3Department of Computing, Mathematical and Statistical Sciences, University of Namibia, Windhoek, Namibia
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.

Author(s): Rikio Ichishima1, Francesc-Antoni Muntaner-Batle2
1Department of Sport and Physical Education, Faculty of Physical Education, Kokushikan University, 7-3-1 Nagayama, Tama-shi, Tokyo 206-8515, Japan.
2Graph 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.
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.

Author(s): Tricia Muldoon Brown1
1Department of Mathematical Sciences, Georgia Southern University, Savannah, GA, U.S.A.
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.