scholarly journals Estimates for the Norm of Generalized Maximal Operator on Strong Product of Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zaryab Hussain ◽  
Ghulam Murtaza ◽  
Toqeer Mahmood ◽  
Jia-Bao Liu
Keyword(s):  
Maximal Operator ◽  
Upper Bounds ◽  
Complete Graphs ◽  
Lower And Upper Bounds ◽  
Connected Graphs ◽  
Strong Product ◽  
Arbitrary Graphs ◽  
Increasing Function ◽  
Product Of Graphs

Let G = G 1 × G 2 × ⋯ × G m be the strong product of simple, finite connected graphs, and let ϕ : ℕ ⟶ 0 , ∞ be an increasing function. We consider the action of generalized maximal operator M G ϕ on ℓ p spaces. We determine the exact value of ℓ p -quasi-norm of M G ϕ for the case when G is strong product of complete graphs, where 0 < p ≤ 1 . However, lower and upper bounds of ℓ p -norm have been determined when 1 < p < ∞ . Finally, we computed the lower and upper bounds of M G ϕ p when G is strong product of arbitrary graphs, where 0 < p ≤ 1 .

scholarly journals On generalized 3-connectivity of the strong product of graphs

2018 ◽  
Vol 12 (2) ◽  
pp. 297-317
Author(s):  
Encarnación Abajo ◽  
Rocío Casablanca ◽  
Ana Diánez ◽  
Pedro García-Vázquez
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Sharp Bound ◽  
Connected Graphs ◽  
Strong Product ◽  
Generalized Connectivity ◽  
Internally Disjoint Trees ◽  
Product Of Graphs

Let G be a connected graph with n vertices and let k be an integer such that 2 ? k ? n. The generalized connectivity kk(G) of G is the greatest positive integer l for which G contains at least l internally disjoint trees connecting S for any set S ? V (G) of k vertices. We focus on the generalized connectivity of the strong product G1 _ G2 of connected graphs G1 and G2 with at least three vertices and girth at least five, and we prove the sharp bound k3(G1 _ G2) ? k3(G1)_3(G2) + k3(G1) + k3(G2)-1.

Product version of reciprocal Gutman indices of composite graphs

2017 ◽  
Vol 26 (2) ◽  
pp. 211-219
Author(s):  
K. Pattabiraman
Keyword(s):  
Tensor Product ◽  
Upper Bounds ◽  
Graph Invariants ◽  
Harary Index ◽  
Connected Graphs ◽  
Zagreb Indices ◽  
Strong Product

In this paper, we present the upper bounds for the product version of reciprocal Gutman indices of the tensor product, join and strong product of two connected graphs in terms of other graph invariants including the Harary index and Zagreb indices.

scholarly journals Removing Twins in Graphs to Break Symmetries

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1111
Author(s):  
Antonio González ◽  
María Luz Puertas
Keyword(s):  
Automorphism Groups ◽  
Upper Bounds ◽  
Unit Interval ◽  
Interval Graphs ◽  
Minimum Size ◽  
Lower And Upper Bounds ◽  
Graph Families ◽  
Arbitrary Graphs

Determining vertex subsets are known tools to provide information about automorphism groups of graphs and, consequently about symmetries of graphs. In this paper, we provide both lower and upper bounds of the minimum size of such vertex subsets, called the determining number of the graph. These bounds, which are performed for arbitrary graphs, allow us to compute the determining number in two different graph families such are cographs and unit interval graphs.

scholarly journals Forcing Subsets for γ∗ tpw -sets in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 451-470
Author(s):  
Cris Laquibla Armada
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Complete Graphs ◽  
Lower And Upper Bounds ◽  
Power Domination ◽  
Wheel Graphs

In this paper, the lower and upper bounds of the forcing total dr-power dominationnumber of any graph are determined. Total dr-power domination number of some special graphs such as complete graphs, star, fan and wheel graphs are shown. Moreover, the forcing total dr-power domination number of these graphs, together with paths and cycles, are determined.

scholarly journals On the Distinguishing Number of Functigraphs

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 332 ◽  
Author(s):  
Muhammad Fazil ◽  
Muhammad Murtaza ◽  
Zafar Ullah ◽  
Usman Ali ◽  
Imran Javaid
Keyword(s):  
Upper Bounds ◽  
Complete Graphs ◽  
Lower And Upper Bounds ◽  
Distinguishing Number ◽  
Vertex Set

Let G 1 and G 2 be disjoint copies of a graph G and g : V ( G 1 ) → V ( G 2 ) be a function. A functigraph F G consists of the vertex set V ( G 1 ) ∪ V ( G 2 ) and the edge set E ( G 1 ) ∪ E ( G 2 ) ∪ { u v : g ( u ) = v } . In this paper, we extend the study of distinguishing numbers of a graph to its functigraph. We discuss the behavior of distinguishing number in passing from G to F G and find its sharp lower and upper bounds. We also discuss the distinguishing number of functigraphs of complete graphs and join graphs.

A short note on inverse sum indeg index of graphs

2019 ◽  
pp. 2050152
Author(s):  
Selvaraj Balachandran ◽  
Suresh Elumalai ◽  
Toufik Mansour
Keyword(s):  
Short Note ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Zagreb Index ◽  
Connected Graphs ◽  
Harmonic Index ◽  
Second Zagreb Index

The inverse sum indeg index of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of the vertex [Formula: see text]. In a recent paper, Pattabiraman [Inverse sum indeg index of graphs, AKCE Int. J. Graphs Combinat. 15(2) (2018) 155–167] gave some lower and upper bounds on [Formula: see text] index of all connected graphs in terms of Harmonic index, second Zagreb index and hyper Zagreb index. But some results were erroneous. In this note, we have corrected these results.

scholarly journals Computing the (k-)monopoly number of direct product of graphs

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1163-1171
Author(s):  
Dorota Kuziak ◽  
Iztok Peterin ◽  
Ismael Yero
Keyword(s):  
Direct Product ◽  
Minimum Degree ◽  
Upper Bounds ◽  
Simple Graph ◽  
Lower And Upper Bounds ◽  
Minimum Cardinality ◽  
Product Graphs ◽  
Product Of Graphs

Let G = (V,E) be a simple graph without isolated vertices and minimum degree ?(G), and let k ? {1-??(G)/2? ,..., ?(G)/2c?} be an integer. Given a set M ? V, a vertex v of G is said to be k-controlled by M if ?M(v)? ?(v)/2 + k where ?M(v) represents the quantity of neighbors v has in M and ?(v) the degree of v. The set M is called a k-monopoly if it k-controls every vertex v of G. The minimum cardinality of any k-monopoly is the k-monopoly number of G. In this article we study the k-monopoly number of direct product graphs. Specifically we obtain tight lower and upper bounds for the k-monopoly number of direct product graphs in terms of the k-monopoly numbers of its factors. Moreover, we compute the exact value for the k-monopoly number of several families of direct product graphs.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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