scholarly journals Topological Indices and f-Polynomials on Some Graph Products

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 292
Author(s):  
Ricardo Abreu-Blaya ◽  
Sergio Bermudo ◽  
José M. Rodríguez ◽  
Eva Tourís
Keyword(s):  
Cartesian Product ◽  
Topological Indices ◽  
Graph Products ◽  
Lexicographic Product ◽  
Inverse Degree

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices.

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.

THE DIAMETER VARIABILITY OF THE CARTESIAN PRODUCT OF GRAPHS

2014 ◽  
Vol 06 (01) ◽  
pp. 1450001 ◽  
Author(s):  
M. R. CHITHRA ◽  
A. VIJAYAKUMAR
Keyword(s):  
Interconnection Networks ◽  
Cartesian Product ◽  
Graph Products ◽  
Single Edge ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

The diameter of a graph can be affected by the addition or deletion of edges. In this paper, we examine the Cartesian product of graphs whose diameter increases (decreases) by the deletion (addition) of a single edge. The problems of minimality and maximality of the Cartesian product of graphs with respect to its diameter are also solved. These problems are motivated by the fact that most of the interconnection networks are graph products and a good network must be hard to disrupt and the transmissions must remain connected even if some vertices or edges fail.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

2-Distance chromatic number of some graph products

2020 ◽  
Vol 12 (02) ◽  
pp. 2050021
Author(s):  
Ghazale Ghazi ◽  
Freydoon Rahbarnia ◽  
Mostafa Tavakoli
Keyword(s):  
Chromatic Number ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Graph Products ◽  
Lexicographic Product ◽  
Graph Product ◽  
Minimum Number

This paper studies the 2-distance chromatic number of some graph product. A coloring of [Formula: see text] is 2-distance if any two vertices at distance at most two from each other get different colors. The minimum number of colors in the 2-distance coloring of [Formula: see text] is the 2-distance chromatic number and denoted by [Formula: see text]. In this paper, we obtain some upper and lower bounds for the 2-distance chromatic number of the rooted product, generalized rooted product, hierarchical product and we determine exact value for the 2-distance chromatic number of the lexicographic product.

scholarly journals On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs

2017 ◽  
Vol 2017 (1) ◽  
Author(s):  
Muhammad Imran ◽  
Shakila Baby ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Muhammad Kashif Shafiq
Keyword(s):  
Cartesian Product ◽  
Topological Indices ◽  
Connected Graphs

On the b-chromatic number of some graph products

2012 ◽  
Vol 49 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Marko Jakovac ◽  
Iztok Peterin
Keyword(s):  
Chromatic Number ◽  
Upper And Lower Bounds ◽  
Vertex Coloring ◽  
Graph Products ◽  
Lexicographic Product ◽  
Arbitrary Graph ◽  
Strong Product ◽  
Color Class ◽  
Complete Bipartite Graphs ◽  
Proper Vertex

A b-coloring is a proper vertex coloring of a graph such that each color class contains a vertex that has a neighbor in all other color classes and the b-chromatic number is the largest integer φ(G) for which a graph has a b-coloring with φ(G) colors. We determine some upper and lower bounds for the b-chromatic number of the strong product G ⊠ H, the lexicographic product G[H] and the direct product G × H and give some exact values for products of paths, cycles, stars, and complete bipartite graphs. We also show that the b-chromatic number of Pn ⊠ H, Cn ⊠ H, Pn[H], Cn[H], and Km,n[H] can be determined for an arbitrary graph H, when integers m and n are large enough.

scholarly journals Edge Regular Graph Products

10.37236/2817 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Boštjan Frelih ◽  
Štefko Miklavič
Keyword(s):  
Regular Graph ◽  
Regular Graphs ◽  
Graph Products ◽  
Lexicographic Product ◽  
Normal Product ◽  
Strong Product

A regular nonempty graph $\Gamma$ is called edge regular, whenever there exists a nonegative integer $\lambda_{\Gamma}$, such that any two adjacent vertices of $\Gamma$ have precisely $\lambda_{\Gamma}$ common neighbours. An edge regular graph $\Gamma$ with at least one pair of vertices at distance 2 is called amply regular, whenever there exists a nonegative integer $\mu_{\Gamma}$, such that any two vertices at distance 2 have precisely $\mu_{\Gamma}$ common neighbours. In this paper we classify edge regular graphs, which can be obtained as a strong product, or a lexicographic product, or a deleted lexicographic product, or a co-normal product of two graphs. As a corollary we determine which of these graphs are amply regular.

scholarly journals Three Topological Indices of Two New Variants of Graph Products

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Muhammad Bilal ◽  
Muhammad Kamran Jamil ◽  
Muhammad Waheed ◽  
Abdu Alameri
Keyword(s):  
Social Sciences ◽  
Complex Network ◽  
New Products ◽  
Topological Indices ◽  
Network Structures ◽  
Graph Products ◽  
Zagreb Index ◽  
Simple Graphs ◽  
Zagreb Indices ◽  
Graph Operation

Graph operations play an important role to constructing complex network structures from simple graphs, and these complex networks play vital roles in different fields such as computer science, chemistry, and social sciences. Computation of topological indices of these complex network structures via graph operation is an important task. In this study, we defined two new variants of graph products, namely, corona join and subdivision vertex join products and investigated exact expressions of the first and second Zagreb indices and first reformulated Zagreb index for these new products.

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 THE WIENER INDEX OF F_H SUMS OF GRAPHS

2021 ◽  
Vol 3 (2) ◽  
pp. 37-57
Author(s):  
L. Alex ◽  
Indulal G
Keyword(s):  
Complete Graph ◽  
Cartesian Product ◽  
Chemical Properties ◽  
Wiener Index ◽  
Topological Indices ◽  
Single Vertex ◽  
Molecular Graphs ◽  
New Class ◽  
And Chemical Properties

Wiener index is the first among the long list of topological indices which was used to correlate structural and chemical properties of molecular graphs. In \cite{Eli} M. Eliasi, B. Taeri defined four new sums of graphs based on the subdivision of edges with regard to the cartesian product and computed their Wiener index. In this paper, we define a new class of sums called $F_H$ sums and compute the Wiener index of the resulting graph in terms of the Wiener indices of the component graphs so that the results in \cite{Eli} becomes a particular case of the Wiener index of $F_H$ sums for $H = K_1$, the complete graph on a single vertex.

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