scholarly journals Edge irregular reflexive labeling for the $ r $-th power of the path

2021 ◽  
Vol 6 (10) ◽  
pp. 10405-10430
Author(s):  
Mohamed Basher ◽  
Keyword(s):  
Natural Number ◽  
Edge Strength ◽  
Vertex Set

<abstract><p>Let $ G(V, E) $ be a graph, where $ V(G) $ is the vertex set and $ E(G) $ is the edge set. Let $ k $ be a natural number, a total k-labeling $ \varphi:V(G)\bigcup E(G)\rightarrow \{0, 1, 2, 3, ..., k\} $ is called an edge irregular reflexive $ k $-labeling if the vertices of $ G $ are labeled with the set of even numbers from $ \{0, 1, 2, 3, ..., k\} $ and the edges of $ G $ are labeled with numbers from $ \{1, 2, 3, ..., k\} $ in such a way for every two different edges $ xy $ and $ x^{'}y^{'} $ their weights $ \varphi(x)+\varphi(xy)+\varphi(y) $ and $ \varphi(x^{'})+\varphi(x^{'}y^{'})+\varphi(y^{'}) $ are distinct. The reflexive edge strength of $ G $, $ res(G) $, is defined as the minimum $ k $ for which $ G $ has an edge irregular reflexive $ k $-labeling. In this paper, we determine the exact value of the reflexive edge strength for the $ r $-th power of the path $ P_{n} $, where $ r\geq2 $, $ n\geq r+4 $.</p></abstract>

scholarly journals On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
I. H. Agustin ◽  
M. I. Utoyo ◽  
Dafik ◽  
M. Venkatachalam ◽  
Surahmat
Keyword(s):  
Natural Number ◽  
Finite Graph ◽  
Star Graph ◽  
Pendant Vertex ◽  
Vertex Set

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

scholarly journals The local antimagic on disjoint union of some family graphs

2019 ◽  
Vol 5 (2) ◽  
pp. 69-75
Author(s):  
Marsidi Marsidi ◽  
Ika Hesti Agustin
Keyword(s):  
Lower Bound ◽  
Natural Number ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Undirected Graph ◽  
Chromatic Numbers ◽  
Minimum Number ◽  
Vertex Set ◽  
Proper Vertex ◽  
Vertex Colouring

A graph  in this paper is nontrivial, finite, connected, simple, and undirected. Graph  consists of a vertex set and edge set. Let u,v be two elements in vertex set, and q is the cardinality of edge set in G, a bijective function from the edge set to the first q natural number is called a vertex local antimagic edge labelling if for any two adjacent vertices and , the weight of  is not equal with the weight of , where the weight of  (denoted by ) is the sum of labels of edges that are incident to . Furthermore, any vertex local antimagic edge labelling induces a proper vertex colouring on where  is the colour on the vertex . The vertex local antimagic chromatic number  is the minimum number of colours taken over all colourings induced by vertex local antimagic edge labelling of . In this paper, we discuss about the vertex local antimagic chromatic number on disjoint union of some family graphs, namely path, cycle, star, and friendship, and also determine the lower bound of vertex local antimagic chromatic number of disjoint union graphs. The chromatic numbers of disjoint union graph in this paper attend the lower bound.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Parallel Residues

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Distance Function ◽  
Length Function ◽  
Coxeter System ◽  
Vertex Set ◽  
Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

UNICUT

Alloy Digest ◽  
1953 ◽  
Vol 2 (12) ◽  
Keyword(s):  
High Speed ◽  
High Speed Steel ◽  
Heat Treating ◽  
Edge Strength

Abstract UNICUT is a molybdenum-tungsten type of high-speed steel containing relatively high percentages of carbon and vanadium. It possesses extremely high edge strength at high hardnesses. This datasheet provides information on composition and hardness. It also includes information on forming, heat treating, machining, and joining. Filing Code: TS-14. Producer or source: Cyclops Corporation.

AL TECH D-5

Alloy Digest ◽  
1980 ◽  
Vol 29 (11) ◽  
Keyword(s):  
Fracture Toughness ◽  
Physical Properties ◽  
Abrasion Resistance ◽  
Heat Treating ◽  
High Carbon ◽  
Die Steel ◽  
High Chromium ◽  
Die Steels ◽  
Edge Strength ◽  
Specialty Steel

Abstract AL Tech D-5 is a high-carbon, high-chromium tool and die steel; it contains cobalt (3.30%) and is air-hardening and nondeforming. The cobalt impacts a pronounced secondary hardening effect in tempering and a non-galling effect. AL Tech D-5 has both superior abrasion resistance and greater edge strength than ordinary high-carbon, high-chromium die steels. It is used for many types of dies, hot-piercing punches and shear blades. This datasheet provides information on composition, physical properties, and hardness as well as fracture toughness. It also includes information on forming, heat treating, and machining. Filing Code: TS-372. Producer or source: AL Tech Specialty Steel Corporation.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

The Natural Numbers

2018 ◽  
Author(s):  
Øystein Linnebo
Keyword(s):  
Natural Number ◽  
Peano Arithmetic ◽  
Number Sequence ◽  
Natural Numbers ◽  
Ordinal Numbers ◽  
Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

scholarly journals Superpixel Segmentation Based on Anisotropic Edge Strength

2019 ◽  
Vol 5 (6) ◽  
pp. 57 ◽  
Author(s):  
Gang Wang ◽  
Bernard De Baets
Keyword(s):  
Distance Measure ◽  
Saliency Detection ◽  
Experimental Results ◽  
Segmentation Method ◽  
Superpixel Segmentation ◽  
Gaussian Kernels ◽  
First Derivative ◽  
Edge Strength

Superpixel segmentation can benefit from the use of an appropriate method to measure edge strength. In this paper, we present such a method based on the first derivative of anisotropic Gaussian kernels. The kernels can capture the position, direction, prominence, and scale of the edge to be detected. We incorporate the anisotropic edge strength into the distance measure between neighboring superpixels, thereby improving the performance of an existing graph-based superpixel segmentation method. Experimental results validate the superiority of our method in generating superpixels over the competing methods. It is also illustrated that the proposed superpixel segmentation method can facilitate subsequent saliency detection.

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