On elementary equivalence of some unoids and unoids of their subsets

В данной работе исследуются свойства уноидов, которые содержат единственную разнозначную функцию. Устанавливаются необходимые и достаточные условия для того, чтобы два таких уноида были элементарно эквивалентными. С помощью этого результата доказываются необходимые и достаточные условия для того, чтобы уноид всех подмножеств уноида $\gA$ был элементарно эквивалентен исходному уноиду $\gA$. In this paper we study the properties of unoids which contain a single injective function. Necessary and sufficient conditions are established for two such unoids to be elementarily equivalent. From this result we obtain necessary and sufficient conditions for the unoid of all subsets of unoid $\gA$ to be elementarily equivalent to the original unoid $\gA$.

k-super cube root cube mean labeling of graphs

Consider a graph G with |V (G)| = p and |E(G)| = q and let f : V (G) → {k, k + 1, k + 2, . . . p + q + k − 1}} be an injective function. The induced edge labeling f ∗ for a vertex labeling f is defined by f ∗ (e) = for all e = uv ∈ E(G) is bijective. If f(V (G)) ∪ {f ∗ (e) : e ∈ E(G)} = {k, k + 1, k + 2, . . . , p + q + k − 1}, then f is called a k-super cube root cube mean labeling. If such labeling exists, then G is a k-super cube root cube mean graph. In this paper, I introduce k-super cube root cube mean labeling and prove the existence of this labeling to the graphs viz., triangular snake graph Tn, double triangular snake graph D(Tn), Quadrilateral snake graph Qn, double quadrilateral snake graph D(Qn), alternate triangular snake graph A(Tn), alternate double triangular snake graph AD(Tn), alternate quadrilateral snake graph A(Qn), & alternate double quadrilateral snake graph AD(Qn).

k-Zumkeller labeling of super subdivision of some graphs

A simple graph $$G=(V,E)$$ G = ( V , E ) is said to be k-Zumkeller graph if there is an injective function f from the vertices of G to the natural numbers N such that when each edge $$xy\in E$$ x y ∈ E is assigned the label f(x)f(y), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we show that the super subdivision of path, cycle, comb, ladder, crown, circular ladder, planar grid and prism are k-Zumkeller graphs.

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.

Expansions of abelian square-free groups

We investigate finitary functions from [Formula: see text] to [Formula: see text] for a square-free number [Formula: see text]. We show that the lattice of all clones on the square-free set [Formula: see text] which contain the addition of [Formula: see text] is finite. We provide an upper bound for the cardinality of this lattice through an injective function to the direct product of the lattices of all [Formula: see text]-linearly closed clonoids, [Formula: see text], to the [Formula: see text] power, where [Formula: see text]. These lattices are studied in [S. Fioravanti, Closed sets of finitary functions between products of finite fields of pair-wise coprime order, preprint (2020), arXiv:2009.02237 ] and there we can find an upper bound for their cardinality. Furthermore, we prove that these clones can be generated by a set of functions of arity at most [Formula: see text].

On ideal sumset labelled graphs

The sumset of two sets A and B of integers, denoted by A + B, is defined as A+B = {a+b : a ∈ A, b ∈ B}. Let X be a non-empty set of non-negative integers. A sumset labelling of a graph G is an injective function f : V (G) → P(X) − {∅} such that the induced function f+ : E(G) → P(X)−{∅} is defined by f+(uv) = f(u) +f(v) ∀uv ∈ E(G). In this paper, we introduce the notion of ideal sumset labelling of graph and discuss the admissibility of this labelling by certain graph classes and discuss some structural characterization of those 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

The monotonicity of Darboux and 2-injective functions

The aim of this work is to establish the monotonicity of a Darboux, 2-injective function

On even vertex odd mean labeling of the calendula graphs

A graph G with |E(G)| = q, an injective function f : V (G) → {0, 2, 4, ..., 2q} is an even vertex odd mean labeling of G that induces the values f(u)+f(v) 2 for the q pairs of adjacent vertices u, v are distinct. In this paper, we investigate an even vertex labeling for the calendula graphs. Moreover we introduce the definition of arbitrary calendula graph and prove that the arbitrary calendula graphs are also even vertex odd mean graphs.

An Odd-Even Sum Labeling of Jellyfish and Mushroom Graphs

A graph G(V,E) with p vertices and q edges called graph odd-even sum if there exists an injective function f from V to {+ 1, +2, +3, ..., +(2p-1)} such that induced a bijection f*(uv)=f(u)+f(v) as label of edge and u,v element of V forms the set {2,4,...,2q}, and f is called odd-even sum labeling. There are three criteria of graphs that can be labeled by this labeling, they are undirected, no loops, and finite for every edges and vertex. Jellyfish J(m,n) graph and Mushroom Mr(m) graph have the criteria. So in this paper will be showed that the Jellyfish and Mushroom graphs can be labeled by this labeling.Keywords: odd-even sum graph; odd-even sum labeling; Jellyfish and mushroom graphs. AbstrakGraf G(V,E) dengan banyak titik p dan sisi q dikatakan graf jumlah ganjil-genap jika terdapat suatu fungsi injetif f dari V ke {+ 1, +2, +3, ..., +(2p-1)} sehingga bijektif f*(uv)=f(u)+f(v) merupakan label sisi dengan u,v anggota dari V membentuk himpunan bilangan {2,4,...,2q}, dengan f merupakan pelabelan jumlah ganjil-genap. Kriteria graf yang dapat dilabeli oleh pelabelan jumlah ganjil-genap ada tiga, yaitu graf yang tidak berarah, tidak memiliki loop, dan terhingga, baik secara sisi maupun titik. Graf Jellyfish J(m,n) dan Mushroom Mr(m) memenuhi ketiga kriteria tersebut. Pada tulisan ini akan ditunjukkan bahwa kedua graf tersebut dapat dilabeli dengan pelabelan jumlah ganjil-genap.Keywords: graf jumlah ganjil-genap; pelabelan jumlah ganjil-genap; graf Jellyfish dan graf Mushroom.