scholarly journals Total Proper Connection and Graph Operations

2021 ◽  
pp. 2142006
Author(s):  
Yingying Zhang ◽  
Xiaoyu Zhu
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Internal Vertex ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Permutation Graphs ◽  
Connection Number ◽  
Graph Operations ◽  
Strong Product ◽  
Proper Connection Number

A graph is said to be total-colored if all the edges and vertices of the graph are colored. A path in a total-colored graph is a total proper path if (i) any two adjacent edges on the path differ in color, (ii) any two internal adjacent vertices on the path differ in color, and (iii) any internal vertex of the path differs in color from its incident edges on the path. A total-colored graph is called total-proper connected if any two vertices of the graph are connected by a total proper path of the graph. For a connected graph [Formula: see text], the total proper connection number of [Formula: see text], denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] total-proper connected. In this paper, we study the total proper connection number for the graph operations. We find that 3 is the total proper connection number for the join, the lexicographic product and the strong product of nearly all graphs. Besides, we study three kinds of graphs with one factor to be traceable for the Cartesian product as well as the permutation graphs of the star and traceable graphs. The values of the total proper connection number for these graphs are all [Formula: see text].

scholarly journals Proper Vertex Connection and Graph Operations

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001
Author(s):  
YINGYING ZHANG ◽  
XIAOYU ZHU
Keyword(s):  
Direct Product ◽  
Connected Graph ◽  
Upper Bounds ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Connection Number ◽  
Graph Operations ◽  
Proper Vertex

A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs. Then we apply these results to some instances of Cartesian and lexicographic product networks.

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.

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 The Reciprocal Complementary Wiener Number of Graph Operations

2021 ◽  
Vol 45 (01) ◽  
pp. 139-154
Author(s):  
R. NASIRI ◽  
A. NAKHAEI ◽  
A. R. SHOJAEIFARD
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Symmetric Difference ◽  
Graph Operations ◽  
Strong Product ◽  
Product Composition ◽  
Wiener Number ◽  
Corona Product

The reciprocal complementary Wiener number of a connected graph G is defined as ∑ {x,y}⊆V (G) 1 D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

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 Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations

2016 ◽  
Vol 17 ◽  
pp. 10-16
Author(s):  
Bommanahal Basavanagoud ◽  
Shreekant Patil
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Upper Bounds ◽  
Zagreb Index ◽  
Graph Operations ◽  
Product Composition ◽  
Degree Of A Vertex ◽  
Product Of Graphs ◽  
Corona Product

The modified second multiplicative Zagreb index of a connected graph G, denoted by $\prod_{2}^{*}(G)$, is defined as $\prod_{2}^{*}(G)=\prod \limits_{uv\in E(G)}[d_{G}(u)+d_{G}(v)]^{[d_{G}(u)+d_{G}(v)]}$ where $d_{G}(z)$ is the degree of a vertex z in G. In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived.The modified second multiplicative Zagreb index of aconnected graph , denoted by , is defined as where is the degree of avertex in . In this paper, we present some upper bounds for themodified second multiplicative Zagreb index of graph operations such as union,join, Cartesian product, composition and corona product of graphs are derived.

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 Sharp Bounds for the Inverse Sum Indeg Index of Graph Operations

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Anam Rani ◽  
Muhammad Imran ◽  
Usman Ali
Keyword(s):  
Tensor Product ◽  
Physicochemical Properties ◽  
Cartesian Product ◽  
Total Surface Area ◽  
Sharp Bounds ◽  
Graph Operations ◽  
Strong Product ◽  
Octane Isomers ◽  
International Academy ◽  
Corona Product

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by ISI G k of G k is a notable predictor of total surface area for octane isomers and is presented as ISI G k = ∑ g k g k ′ ∈ E G k d G k g k d G k g k ′ / d G k g k + d G k g k ′ , where d G k g k represents the degree of g k ∈ V G k . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.

Maximum value of conflict-free vertex-connection number of graphs

2018 ◽  
Vol 10 (05) ◽  
pp. 1850059 ◽  
Author(s):  
Zhenzhen Li ◽  
Baoyindureng Wu
Keyword(s):  
Free Path ◽  
Connected Graph ◽  
Colored Graph ◽  
Connection Number ◽  
Free Vertex ◽  
Maximum Value

A path in a vertex-colored graph is called conflict-free if there is a color used on exactly one of its vertices. A vertex-colored graph is said to be conflict-free vertex-connected if any two vertices of the graph are connected by a conflict-free path. The conflict-free vertex-connection number, denoted by [Formula: see text], is defined as the smallest number of colors required to make [Formula: see text] conflict-free vertex-connected. Li et al. [Conflict-free vertex-connections of graphs, preprint (2017), arXiv:1705.07270v1[math.CO]] conjectured that for a connected graph [Formula: see text] of order [Formula: see text], [Formula: see text]. We confirm that the conjecture is true and poses two relevant conjectures.

scholarly journals The forgotten index of complement graph operations and its applications of molecular graph

2020 ◽  
Vol 3 (3) ◽  
pp. 53-61
Author(s):  
Mohammed Saad Alsharafi ◽  
◽  
Mahioub Mohammed Shubatah ◽  
Abdu Qaid Alameri ◽  
◽  
...  
Keyword(s):  
Molecular Graph ◽  
Cartesian Product ◽  
Numerical Parameter ◽  
Graph Operations ◽  
Strong Product ◽  
Complement Graph ◽  
Structure Activity ◽  
Forgotten Index ◽  
Quantitative Structure Activity Relationships ◽  
Mathematical Operations

A topological index of graph \(G\) is a numerical parameter related to graph which characterizes its molecular topology and is usually graph invariant. Topological indices are widely used to determine the correlation between the specific properties of molecules and the biological activity with their configuration in the study of quantitative structure-activity relationships (QSARs). In this paper some basic mathematical operations for the forgotten index of complement graph operations such as join \(\overline {G_1+G_2}\), tensor product \(\overline {G_1 \otimes G_2}\), Cartesian product \(\overline {G_1\times G_2}\), composition \(\overline {G_1\circ G_2}\), strong product \(\overline {G_1\ast G_2}\), disjunction \(\overline {G_1\vee G_2}\) and symmetric difference \(\overline {G_1\oplus G_2}\) will be explained. The results are applied to molecular graph of nanotorus and titania nanotubes.

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