Resulting Edge Labels Research Articles

ABSOLUTE MEAN GRACEFUL LABELING IN THE CONTEXT OF m-SPLITTING AND DEGREE SPLITTING GRAPHS

A graph G with q edges is said to be absolute mean graceful if there is a one-to-one function f from V (G) to the set {0,±1,±2,±3,...,±q} such that when each edge xy is assigned the label |f(x)−f(y)| 2 , then the resulting edge labels are distinct. In this paper, the absolute mean graceful labeling of m-splitting and degree splitting graphs of some graphs are investigated.

Anti Skolem Mean Labeling of Quardrilateral Snake Related Graphs

A graph with p vertices and q edges where is said to be an anti skolem mean graph if it is possible to label the vertices with distinct labels from in such a way that each edge e=uv is labelled with then the resulting edge labels are distinct labels from the set . In this case f is called an Anti Skolem Mean labelling of G.

A Study on Power-3 Heronian Odd Mean Labeling for some Path Related Graphs

In this article, we discussed Power-3 Heronian odd Mean Labeling for some path related graphs. A function is said to be Power 3 Heronian odd mean labeling of a graph G with q edges, if f is a bijective function from the vertices of G to the set{1,3,5,.......2p-1}such that when each edges uv is assigned the label, then the resulting edge labels are distinct numbers.

PELABELAN ODD-GRACEFUL PADA GRAF PRODUK SISIR

Gnanajothi defined a graph with edges to be odd-graceful if there is an injective function such that if every edge is labelled with the resulting edge labels are . She proved that the graph obtained by joining one pendant to every vertex in is odd-graceful if and only if is even. In this paper, we extend her results by proving that is odd-graceful if and only if is even.Keywords: Vertex-labelling, odd-graceful, cycleMSC 2020: 05C78

K-Zumkeller labeling of super subdivision of some graphs

A simple graph 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 xyin 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.

Root square mean labelling of some graphs obtained from path

Let G be a graph with q edges. A labelling f of G is said to be root square mean labelling if f: V(G)UE(G)→{1,2,…,q+1} such that when each edge e = uv labelled with f(e)=[f(u)2+f(v)2/2] or f(e)=[f(u)2+f(v)2/2] then the resulting edge labels are distinct. A graph G is called a root square mean graph if G can be labelled by a root square mean labelling. In this paper we determine a root mean square labelling of two graphs obtained from path, which are corona product of ladder and complete graph with order 1, and a graph obtained from triangular snake by join one vertex with degree 2 in each triangle to a new vertex. The method of labelling construction is we need to do labeling to the vertices of the graph with label 1, 2, 3, …, q+1. The labels of the vertices are not necessarily different. The next step is we need to do labeling to the edges with the certain formula by using the vertex labeling. The edge labels must be different. By the labelling we construct, we proof that the two graphs are root square mean graphs.

An application of super mean and magic graphs labeling on cryptography system

An edge-magic total (EMT) labeling on a simple graph G with p vertices and q edges is a bijection f: V∪E→{1, 2, …, p + q} such that for each edge xy of G, f(x) + f(xy) + f(y) = k, for a fixed positive integer k. Moreover, an EMT labeling f is a super edge-magic total labeling (SEMT) if f(V) = {1, 2, …, p}. Furthermore, a graph G is called a super mean (SM) graph if there exists an injective function f from the vertices of G to {0,1,2,…,q} such that when each edge uv is labeled with (f(u) +f(v))/2 when f(u) +f(v) is even, and (f(u) +f(v) + 1)/2 when f(u) +f(v) is odd, then the resulting edge labels are distinct. There are two essentials results that will be proposed in this paper. In the first result, we show that a graph (n, 1) − F Caterpillar has a SM labeling and a graph Hu,y has a SEMT labeling. In the second result, we will give an application of these labeling to increase the security level of Affine Cipher in which to encrypt a text on socials media.

K-Zumkeller labeling of the Cartesian and tensor product of paths and cycles

A k-Zumkeller labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that each edge uv ∈ E is assigned the label f (u) f (v), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we prove that the graph Pm × Pn is k-Zumkeller graph for m, n ≥ 3 while Pm × Cn and Cm × Cn are k-Zumkeller graphs for n ≡ 4 (mod2). Also we show that the graphs Pm ⊗ Pn and Pm ⊗ Cn for m, n ≥ 3 admit k-Zumkeller labeling. Further, the graph Cm ⊗ Cn where m or n is even admit a k-Zumkeller labeling.

Some general results on restricted k-mean graph

A graph G = (p, q) is said to have a restricted k-mean labeling if there is an injective function f from the vertices of G to {k – 1, k, k + 1, … , k + q – 1} such that when each edge uv is labeled with then the resulting edge labels {k, k + 1, k + 2, … , k + q – 1} are all distinct where k is a positive integer greater than or equal to one. A graph that admits a restricted k-mean labeling is called a restricted k-mean graph.

New classes of mean graphs

A graph G = (V, E) with p vertices and q edges is said to be a mean graph if it is possible to label the vertices x ϵ V with distinct elements f (x) from 0, 1, 2, … , q in such a way that when each edge e = uv is labeled with if f (u) + f (v) is even and if f (u) + f (v) is odd, then the resulting edge labels are distinct. In this case f is called a mean labeling of G. In this paper we prove that Triangular Ladder TLn, TLn, ʘ K1, Tn, ʘ K1, D(Tn) ʘ K1, Qn ʘ K1, D(Qn) ʘ K1 are mean graphs.

Prime Graceful Labeling

A graph G with m vertices and n edges, is said to be prime graceful labeling, if there is an injection from the vertices of G to {1, 2, ..., k} where k = min {2m, 2n} such that gcd ( ( ), ( )=1 and the induced injective function from the edges of G to {1, 2, ..., k − 1} defined by ( ) = | ( ) − ( ) | , the resulting edge labels are distinct. In this paper path , cycle Cn , star K1,n , friendship graph Fn , bistar Bn,n, C4 ∪ Pn , Km,2 and Km,2 ∪ Pn are shown to be Prime Graceful Labeling .

Skolem Difference Fibonacci Mean Labeling of H Class of Graphs

A graph G with p vertices and q edges is said to have Skolem difference Fibonacci mean labeling if it is possible to label the vertices x 15∈> V with distinct elements f ( x ) from the set { 1,2 ,..., F p+q } in such a way that the edge e = uv is labelled with 15 fu-f(v)2> if 15fu-f(v)> is even and if is odd and the resulting edge labels are distinct and are from Fibonacci numbers { F 1 , F 2 ,...,F q }, where F 1 =1 and F 2 = 2 . A graph that admits Skolem difference Fibonacci mean labeling is called a Skolem difference Fibonacci mean graph. In this paper, we prove that H-class of graphs are Skolem difference Fibonacci mean graphs.

Elegant labeling of some graphs and their line graphs

An elegant labeling f of graph G with ‘ q ’ edges an injective function from the vertices of G to the set {0, 1 , 2,------- q} such that when each edge ( e = xy ) is assigned the label { ( f (x) + f (y) ) mod( q + 1) } the resulting edge labels are distinct and non zero. In this paper it is shown to be certain families of line graphs of elegant graphs are elegant graphs.

Generating Graceful Trees from Caterpillars by Recursive Attachment

A graceful labeling of a graph G with n edges is an injection f:V(G)→{0,1,2,⋯,n} with the property that the resulting edge labels are also distinct, where an edge incident with vertices u and v is assigned the label |f(u)−f(v)|. A graph which admits a graceful labeling is called a graceful graph. In this paper, inspired by Koh [K.M. Koh, D.G. Rogers and T. Tan, Two theorems on graceful trees, Discrete Math., 25 (1979), 141–148] method, which combines a known graceful trees to obtain a larger graceful trees, we introduced a new method of combining graceful trees called recursive attachment method, and we show that the recursively attached tree Ti=Ti−1⊕TAi−1 is graceful, for i≥1, where T0 is a base tree which is taken as a caterpillar and TAi−1 is an attachment tree which taken as any caterpillar. Here Ti−1⊕TAi−1 represents a tree obtained by attaching a copy of TAi−1 at each vertex of degree at least two in Ti−1, for i≥1. Consequently the graceful tree conjecture is true for every recursively attached caterpillar tree Ti, for i≥1.

Any Tree with [formula omitted] edges can be embedded in a Graceful Tree with less than [formula omitted] edges and in a graceful planar graph

A function f is called a graceful labeling of a graph G with m edges, if f is an injective function from V(G) to {0,1,2,…,m} such that when every edge uv is assigned the edge label |f(u)−f(v)|, then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. In this paper, we prove a basic structural property of graceful graphs, that every tree can be embedded as a spanning subtree in a graceful planar graph. Also we show that any tree with m edges can be embedded in a graceful tree with less than 4m edges. A range-relaxed graceful labeling f is defined from V(G) to 0,1,2,…,m′, where m′≥m. We improve the bound 2m−diam(T) on the range-relaxed graceful labeling given by Van Bussel (2002) for a tree T.

Geometric mean labeling on Degree splitting graphs

A Graph G = (V, E) with p vertices and q edges is said to be a Geometric mean graph if it is possible to label the vertices xϵV with distinct labels f(x) from 1, 2 … … … … q + 1 in such a way that when each edge e = uv is labeled with f (e = uv) = ⌈ f (u) f (v)⌉ or ⌊ f (u) f (v)⌋, then the resulting edge labels are distinct. In this case, f is called Geometric mean labeling of G. In this paper we study the Geometric mean labeling on Degree Splitting graphs.

Even harmonious labelings of disjoint graphs with a small component

A graph G with q edges is said to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y)(modq), the resulting edge labels are distinct. If G is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2(q−1) and the induced function f∗ from the edges of G to 0,2,…,2(q−1) defined by f∗(xy)=f(x)+f(y)(mod2q) is bijective. We investigate the existence of properly even harmonious labelings of families of disconnected graphs with one of C3,C4,K4 or W4 as a component.

Properly even harmonious labelings of disconnected graphs

A graph G with q edges is said to be harmonious if there is an injection f from the vertices of G to the group of integers modulo q such that when each edge xy is assigned the label f(x)+f(y)(modq), the resulting edge labels are distinct. If G is a tree, exactly one label may be used on two vertices. Over the years, many variations of harmonious labelings have been introduced.We study a variant of harmonious labeling. A function f is said to be a properly even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2(q−1) and the induced function f∗ from the edges of G to 0,2,…,2(q−1) defined by f∗(xy)=f(x)+f(y)(mod2q) is bijective. This paper focuses on the existence of properly even harmonious labelings of the disjoint union of cycles and stars, unions of cycles with paths, unions of squares of paths, and unions of paths.

Embedding an Arbitrary Tree in a Graceful Tree

A function f is called a graceful labeling of a graph G with m edges if f is an injective function from V(G) to \(\{0,1,2,\ldots ,m\}\) such that when every edge uv is assigned the edge label \(|f(u)-f(v)|\), then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular Graceful Tree Conjecture states that every tree is graceful. The Graceful Tree Conjecture remains open for over four decades. Although there are a few general results and techniques on the construction of graceful trees, settling the conjecture seems to be very hard. In this paper, we have introduced a new and different method of constructing graceful trees from a given arbitrary tree. More precisely, we show that every tree can be embedded in a graceful tree with at most km edges, \(k<\lceil \frac{m}{4} \rceil \), where m is the number of edges of the given arbitrary tree. Further, we pose a related open problem toward settling the Graceful Tree Conjecture.

Towards Optimal Embedding of an Arbitrary Tree in a Graceful Tree

A function f is called a graceful labeling of a graph G with m edges, if f is an injective function from V(G) to {0,1,2,⋯,m} such that when every edge uv is assigned the edge label |f(u)−f(v)|, then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular and notorious Graceful Tree Conjecture: “All trees are graceful”, remains unsettled over four decades inspite of many rigorous attempts and investigations. Inspired by the interesting result of Acharya et al. [B.D. Acharya, S.B. Rao, S. Arumugam, Embedding and NP-Complete problems for Graceful Graphs, Labelings of Discrete Structures and Applications, in: B.D. Acharya, S. Arumugam, Alexander Rosa (Eds.), Narosa Publishing House, New Delhi, 2008, pp. 57–62], in this paper we show that every tree can be embedded in a graceful tree. Practically, we observe that any tree with m edges can be embedded in a graceful tree with less than 2m edges and we discuss a related open problem towards settling the Graceful Tree Conjecture through the embedding approach.