P-Normality

A topological space X is called P-normal if there exist a normal space Y and a bijective function f : X −→ Y such that the restriction f|A : A −→ f(A) is a homeomorphism for each paracompact subspace A ⊆ X. We will investigate this property and produce some examples to illustrate the relation between P-normality and other weaker kinds of normality.

Prime labeling of line and splitting graph of brush graph

A bijective function f from V(G) to {1,2,…, n} be a prime labeling of a graph G with n order if for every u, v ∈ V(G) such that e = uv ∈ E(G), f(u) and f(v) relatively prime. A prime graph is a graph which admits prime labeling. In this study, we investigate and conclude that the line and splitting graph of the brush graph is a prime graph.

Characterization of the Congestion Lemma on Layout Computation

An embedding of a guest network G N into a host network H N is to find a suitable bijective function between the vertices of the guest and the host such that each link of G N is stretched to a path in H N . The layout measure is attained by counting the length of paths in H N corresponding to the links in G N and with a complexity of finding the best possible function overall graph embedding. This measure can be computed by summing the minimum congestions on each link of H N , called the congestion lemma. In the current study, we discuss and characterize the congestion lemma by considering the regularity and optimality of the guest network. The exact values of the layout are generally hard to find and were known for very restricted combinations of guest and host networks. In this series, we derive the correct layout measures of circulant networks by embedding them into the path- and cycle-of-complete graphs.

Super H -Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

Let G be a graph and H ⊆ G be subgraph of G . The graph G is said to be a , d - H antimagic total graph if there exists a bijective function f : V H ∪ E H ⟶ 1,2,3 , … , V H + E H such that, for all subgraphs isomorphic to H , the total H weights W H = W H = ∑ x ∈ V H f x + ∑ y ∈ E H f y forms an arithmetic sequence a , a + d , a + 2 d , … , a + n − 1 d , where a and d are positive integers and n is the number of subgraphs isomorphic to H . An a , d - H antimagic total labeling f is said to be super if the vertex labels are from the set 1,2 , … , | V G . In this paper, we discuss super a , d - C 3 -antimagic total labeling for generalized antiprism and a super a , d - C 8 -antimagic total labeling for toroidal octagonal map.

Results on C2-paracompactness

A C-paracompact is a topological space X associated with a paracompact space Y and a bijective function f : X −→ Y satisfying that f A: A −→ f(A) is a homeomorphism for each compact subspace A ⊆ X. Furthermore, X is called C2-paracompact if Y is T2 paracompact. In this article, we discuss the above concepts and answer the problem of Arhangel’ski ̆i. Moreover, we prove that the sigma product Σ(0) can not be condensed onto a T2 paracompact space.

Graceful Labeling and Skolem Graceful Labeling on the U-star Graph and It’s Application in Cryptography

Graceful Labeling on graph G=(V, E) is an injective function f from the set of the vertex V(G) to the set of numbers {0,1,2,...,|E(G)|} which induces bijective function f from the set of edges E(G) to the set of numbers {1,2,...,|E(G)|} such that for each edge uv e E(G) with u,v e V(G) in effect f(uv)=|f(u)-f(v)|. Meanwhile, the Skolem graceful labeling is a modification of the Graceful labeling. The graph has graceful labeling or Skolem graceful labeling is called graceful graph or Skolem graceful labeling graph. The graph used in this study is the U-star graph, which is denoted by U(Sn). The purpose of this research is to determine the pattern of the graceful labeling and Skolem graceful labeling on graph U(Sn) apply it to cryptography polyalphabetic cipher. The research begins by forming a graph U(Sn) and they are labeling it with graceful labeling and Skolem graceful labeling. Then, the labeling results are applied to the cryptographic polyalphabetic cipher. In this study, it is found that the U(Sn) graph is a graceful graph and a Skolem graceful graph, and the labeling pattern is obtained. Besides, the labeling results on a graph it U(Sn) can be used to form a table U(Sn) polyalphabetic cipher. The table is used as a key to encrypt messages.

LH LABELING OF GRAPHS

A graph $G$ with $n$ vertices is said to have an LH labeling if there exists a bijective function $f: V(G)$ to $\{1,2,3,\ldots ,n\} $ such that the induced map $f^*: E(G)\rightarrow N$, the set of natural numbers defined by $f^*(uv) = \frac{LCM(f(u), f(v))} {HCF(f(u),f(v))}$ is injective (LCM and HCF denotes the least common multiple and highest common factor respectively). A graph that admits an LH labeling is called an LH graph. This article explores the results of LH Labeling of some standard graphs.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

Bijective model of time and J.T. Fraser’s Nowless Universe

In bijective modelling, the physical reality is represented by the set X, the model of physical reality by the set Y. Every element in the set X has exactly one correspondent element in the set Y. Set X and set X are related by the bijective function f:X→Y. Bijective modelling is confirming that time is the duration of given system entropy increasing in time-invariant space. Time-invariant space is the fundamental arena of the Nowless Universe.

Olbers’ paradox bijective solution

The bijective research of the Olbers’ paradox based on the bijective function of set theory confirms that the paradox is fictitious. The set X is the universe and set Y is the model of the universe. Every element in the set X has exactly one correspondent model in the set Y. Elements in both sets are defined on the basis of the elementary perception. NASA has measured that the universal space has Euclidean shape, which means it is infinite. The luminosity of stars that are on a finite distance from the Earth is not strong enough to make the night a day.