On Edge Irregular Reflexive Labeling of Categorical Product of Two Paths

Among the huge diversity of ideas that show up while studying graph theory,one that has obtained a lot of popularity is the concept of labelings of graphs.Graph labelings give valuable mathematical models for a wide scope of applications in high technologies(cryptography,astronomy,data security,various coding theory problems,communication networks,etc.).A labeling or a valuation of a graph is any mapping that sends a certain set of graph elements to a certain set of numbers subject to certain conditions.Graph labeling is a mapping of elements of the graph,i.e.,vertex and for edges to a set of numbers(usually positive integers),called labels.If the domain is the vertex-set or the edge-set,the labelings are called vertex labelings or edge labelings respectively.Similarly,if the domain is V(G)[E(G)],then the labeling is called total labeling.A reflexive edge irregular k-labeling of graph introduced by Tanna et al.:A total labeling of graph such that for any two different edges ab and a'b'of the graph their weights has wt_(x)(ab)=x(a)+x(ab)+x(b) and wt_(x)(a'b')=x(a')+x(a'b')+x(b') are distinct.The smallest value of k for which such labeling exist is called the reflexive edge strength of the graph and is denoted by res(G).In this paper we have found the exact value of the reflexive edge irregularity strength of the categorical product of two paths (P_(a)×P_(b))for any choice of a≥3 and b≥3.

Edge Product Cordial Labeling of Some Cycle Related Graphs

For a graph having no isolated vertex, a function is called an edge product cordial labeling of graph G, if the induced vertex labeling function defined by the product of labels of incident edges to each vertex is such that the number of edges with label 0 and the number of edges with label 1 differ by at most 1 and the number of vertices with label 0 and the number of vertices with label 1 also differ by at most 1. In this paper, we discuss edge product cordial labeling for some cycle related graphs.

Some Edge Product Cordial Graphs in the Context of Duplication of Some Graph Elements

For a graph, a function is called an edge product cordial labeling of G, if the induced vertex labeling function is defined by the product of the labels of the incident edges as such that the number of edges with label 1 and the number of edges with label 0 differ by at most 1 and the number of vertices with label 1 and the number of vertices with label 0 differ by at most 1. In this paper, we show that the graphs obtained by duplication of a vertex, duplication of a vertex by an edge or duplication of an edge by a vertex in a crown graph are edge product cordial. Moreover, we show that the graph obtained by duplication of each of the vertices of degree three by an edge in a gear graph is edge product cordial. We also show that the graph obtained by duplication of each of the pendent vertices by a new vertex in a helm graph is edge product cordial.

QoS in FANET Business and Swarm Data

This article shows the quality of services in a wireless swarm of drones that form an ad hoc network between them Fly Ad Hoc Networks(FANET).Each drone has the ability to send and receive information(like a router);and can behave as a hierarchical node whit the intregration of three protocols:Multiprotocol Label Switch(MPLS),Fast Hierarchical AD Hoc Mobile(FHAM)and Internet Protocol version 6(IPv6),in conclusion MPLS+FHAM+IPv6.The metrics analyzed in the FANET are:delay,jitter,throughput,lost and sent packets/received.Testing process was carried out with swarms composed of 10,20,30 and 40 units;In this work,the stage with 40 droneswas analyzed showing registration processes,and sentmessages sequences between different drones that were part of the same swarm.A special analysis about the traffic between drones(end-to-end)was carried out,as well as the possible security flaws in each drone and the current status and future trends in real services.Regarding future trends,in a real environment,we took as a starting point,metrics results obtained in the simulation(positive according to the obtained results).These results gave us a clear vision of how the network will behave in a real environment with the aim to carry out the experiment on a physical level in the near future.This work also shows the experience quality from the service quality metrics obtained through a mathematical model.This quality of experience model will allow us to use it objectively in the agricultural sector,which is a great interest area and is where we are working with drones.Finally in this article we show our advances for a business model applied to the aforementioned agricultural sector,as well as the data analysis and services available to the end customer.These services available to the end customer have been classified into a basic,medium,advanced and plus level.

Vertex-antimagic Labelings of Regular Graphs

Let G = （V, E） be a finite, simple and undirected graph with p vertices and q edges. An （a, d）-vertex-antimagic total labeling of G is a bijection f from V（G） t2 E（G） onto the set of consecutive integers 1, 2,... ,p ＋ q, such that the vertex-weights form an arithmetic progression with the initial term a and difference d, where the vertex-weight of x is the sum of the value f（x） assigned to the vertex x together with all values f（xy） assigned to edges xy incident to x. Such labeling is called super if the smallest possible labels appear on the vertices. In this paper, we study the properties of such labelings and examine their existence for 2r-regular graphs when the difference d is 0, 1,..., r ＋ 1.

Strong Labelings of Linear Forests

A （p, q）-graph G is called super edge-magic if there exists a bijective function f ： V（G） U E（G） →{1, 2 p＋q} such that f（u）＋ f（v）＋f（uv） is a constant for each uv C E（G） and f（Y（G）） = {1,2,...,p}. In this paper, we introduce the concept of strong super edge-magic labeling as a particular class of super edge-magic labelings and we use such labelings in order to show that the number of super edge-magic labelings of an odd union of path-like trees （mT）, all of them of the same order, grows at least exponentially with m.