scholarly journals On $ {L}(2,1) $-labelings of some products of oriented cycles

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lucas Colucci ◽  
Ervin Győri
Keyword(s):  
Cartesian Product ◽  
Strong Product

<p style='text-indent:20px;'>We refine two results of Jiang, Shao and Vesel on the <inline-formula><tex-math id="M2">\begin{document}$ L(2,1) $\end{document}</tex-math></inline-formula>-labeling number <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of <inline-formula><tex-math id="M4">\begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ n \geq 40 $\end{document}</tex-math></inline-formula>; in the case of strong product, we either compute the exact value or establish a gap of size one for <inline-formula><tex-math id="M7">\begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n \geq 48 $\end{document}</tex-math></inline-formula>.</p>

scholarly journals The Second Hyper-Zagreb Index of Complement Graphs and Its Applications of Some Nano Structures

2021 ◽  
pp. 54-75
Author(s):  
Mohammed Alsharafi ◽  
Yusuf Zeren ◽  
Abdu Alameri
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Quantitative Structure Activity Relationship ◽  
Quantitative Structure ◽  
Structure Property ◽  
Zagreb Index ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graph Theory ◽  
Mathematical Operations

In chemical graph theory, a topological descriptor is a numerical quantity that is based on the chemical structure of underlying chemical compound. Topological indices play an important role in chemical graph theory especially in the quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR). In this paper, we present explicit formulae for some basic mathematical operations for the second hyper-Zagreb index of complement graph containing the join G1 + G2, tensor product G1 \(\otimes\) G2, Cartesian product G1 x G2, composition G1 \(\circ\) G2, strong product G1 * G2, disjunction G1 V G2 and symmetric difference G1 \(\oplus\) G2. Moreover, we studied the second hyper-Zagreb index for some certain important physicochemical structures such as molecular complement graphs of V-Phenylenic Nanotube V PHX[q, p], V-Phenylenic Nanotorus V PHY [m, n] and Titania Nanotubes TiO2.

scholarly journals The partition dimension of strong product graphs and Cartesian product graphs

2014 ◽  
Vol 331 ◽  
pp. 43-52 ◽  
Author(s):  
Ismael González Yero ◽  
Marko Jakovac ◽  
Dorota Kuziak ◽  
Andrej Taranenko
Keyword(s):  
Cartesian Product ◽  
Strong Product ◽  
Product Graphs ◽  
Partition Dimension ◽  
Cartesian Product Graphs

scholarly journals Interval-Valued Fuzzy Soft Graphs

2017 ◽  
Vol 5 (1) ◽  
pp. 19-27 ◽  
Author(s):  
Onur Zihni ◽  
Yıldıray Çelik ◽  
Güven Kara
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Soft Sets ◽  
Strong Product ◽  
Different Types ◽  
Fuzzy Soft Sets ◽  
Interval Valued

Abstract In this paper, we combine concepts of interval-valued fuzzy soft sets and graph theory. Then we introduce notations of interval-valued fuzzy soft graphs and complete interval-valued fuzzy soft graphs. We also present several different types operations including cartesian product, strong product and composition on interval-valued fuzzy soft graphs and investigate some properties of them.

scholarly journals On the Laplacian spectra of product graphs

2015 ◽  
Vol 9 (1) ◽  
pp. 39-58 ◽  
Author(s):  
S. Barik ◽  
R.B. Bapat ◽  
S. Pati
Keyword(s):  
Cartesian Product ◽  
Complete Characterization ◽  
Graph Products ◽  
Lexicographic Product ◽  
Laplacian Spectrum ◽  
Strong Product ◽  
Laplacian Spectra ◽  
Product Graphs ◽  
Characteristic Sets

Graph products and their structural properties have been studied extensively by many researchers. We investigate the Laplacian eigenvalues and eigenvectors of the product graphs for the four standard products, namely, the Cartesian product, the direct product, the strong product and the lexicographic product. A complete characterization of Laplacian spectrum of the Cartesian product of two graphs has been done by Merris. We give an explicit complete characterization of the Laplacian spectrum of the lexicographic product of two graphs using the Laplacian spectra of the factors. For the other two products, we describe the complete spectrum of the product graphs in some particular cases. We supply some new results relating to the algebraic connectivity of the product graphs. We describe the characteristic sets for the Cartesian product and for the lexicographic product of two graphs. As an application we construct new classes of Laplacian integral graphs.

scholarly journals The Second Hyper-Zagreb Coindex of Chemical Graphs and Some Applications

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ahmed Ayache ◽  
Abdu Alameri ◽  
Mohammed Alsharafi ◽  
Hanan Ahmed
Keyword(s):  
Tensor Product ◽  
Topological Index ◽  
Molecular Graph ◽  
Cartesian Product ◽  
Explicit Formulas ◽  
Chemical Graph ◽  
Strong Product ◽  
Chemical Graphs ◽  
Product Composition ◽  
Graph Structures

The second hyper-Zagreb coindex is an efficient topological index that enables us to describe a molecule from its molecular graph. In this current study, we shall evaluate the second hyper-Zagreb coindex of some chemical graphs. In this study, we compute the value of the second hyper-Zagreb coindex of some chemical graph structures such as sildenafil, aspirin, and nicotine. We also present explicit formulas of the second hyper-Zagreb coindex of any graph that results from some interesting graphical operations such as tensor product, Cartesian product, composition, and strong product, and apply them on a q-multiwalled nanotorus.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

On bipolar fuzzy soft graphs

2018 ◽  
Vol 27 (2) ◽  
pp. 123-132
Author(s):  
Yildiray Çelik ◽  
Keyword(s):  
Graph Theory ◽  
Cartesian Product ◽  
Soft Sets ◽  
Strong Product ◽  
Different Types ◽  
Fuzzy Soft Sets

In this paper, we combine the concepts of bipolar fuzzy soft sets and graph theory. Then we introduce notations of bipolar fuzzy soft graph and strong bipolar fuzzy soft graph. We also present several different types of operations including cartesian product, strong product and composition on bipolar fuzzy soft graphs and investigate some properties of them.

scholarly journals Efficient Open Domination in Digraph Products

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 496
Author(s):  
Dragana Božović ◽  
Iztok Peterin
Keyword(s):  
Cartesian Product ◽  
Lexicographic Product ◽  
Underlying Graph ◽  
Strong Product ◽  
Cyclic Graphs

A digraph D is an efficient open domination digraph if there exists a subset S of V ( D ) for which the open out-neighborhoods centered in the vertices of S form a partition of V ( D ) . In this work we deal with the efficient open domination digraphs among four standard products of digraphs. We present a method for constructing the efficient open domination Cartesian product of digraphs with one fixed factor. In particular, we characterize those for which the first factor has an underlying graph that is a path, a cycle or a star. We also characterize the efficient open domination strong product of digraphs that have factors whose underlying graphs are uni-cyclic graphs. The full characterizations of the efficient open domination direct and lexicographic product of digraphs are also given.

scholarly journals The Sigma Coindex of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yasar Nacaroglu
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Nonadjacent Vertex ◽  
Graph Operations ◽  
Strong Product ◽  
Mathematical Properties ◽  
Corona Product

The sigma coindex is defined as the sum of the squares of the differences between the degrees of all nonadjacent vertex pairs. In this paper, we propose some mathematical properties of the sigma coindex. Later, we present precise results for the sigma coindices of various graph operations such as tensor product, Cartesian product, lexicographic product, disjunction, strong product, union, join, and corona product.

scholarly journals Further results on non-self-centrality measures of graphs

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5137-5148 ◽  
Author(s):  
Mahdieh Azari
Keyword(s):  
Structural Parameters ◽  
Cartesian Product ◽  
Upper And Lower Bounds ◽  
Centrality Measures ◽  
Eccentric Connectivity Index ◽  
The Third ◽  
Eccentricity Index ◽  
Strong Product ◽  
Special Cases ◽  
Special Case

For indicating the non-self-centrality extent of graphs, two new eccentricity-based measures namely third Zagreb eccentricity index E3(G) and non-self-centrality number N(G) of a connected graph G have recently been introduced as E3(G) = ?uv?E(G)|?G(u)-?G(v)| and N(G) = ? {u,v}?V(G) |?G(u)-?G(v)|, where ?G(u) denotes the eccentricity of a vertex u in G. In this paper, we find relation between the third Zagreb eccentricity index of graphs with some eccentricity-based invariants such as second Zagreb eccentricity index and second eccentric connectivity index. We also give sharp upper and lower bounds on the nonself-centrality number of graphs in terms of some structural parameters and relate it to two well-known eccentricity-based invariants namely total eccentricity and first Zagreb eccentricity index. Furthermore, we present exact expressions or sharp upper bounds on the third Zagreb eccentricity index and non-selfcentrality number of several graph operations such as join, disjunction, symmetric difference, lexicographic product, strong product, and generalized hierarchical product. The formulae for Cartesian product and rooted product as two important special cases of generalized hierarchical product and the formulae for corona product as a special case of rooted product are also given.

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