Sum Of Labels Research Articles

An Odd-Even Sum Labeling of Jellyfish and Mushroom Graphs

AbstractA 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.

Local antimagic orientation of graphs

An antimagic labelling of a digraph D with m arcs is a bijection from the set of arcs of D to $$\{1,\ldots ,m\}$$ such that any two vertices have distinct vertex-sums, where the vertex-sum of a vertex $$v\in V(D)$$ is the sum of labels of all arcs entering v minus the sum of labels of all arcs leaving v. An orientation D of a graph G is antimagic if D has an antimagic labelling. In 2010, Hefetz, M$$\ddot{\text {u}}$$tze and Schwartz conjectured that every connected graph admits an antimagic orientation. The conjecture is still open, even for trees. Motivated by directed version of the well-known 1-2-3 Conjecture, we deal with vertex-sums such that only adjacent vertices must be distinguished. An orientation D of a graph G is local antimagic if there is a bijection from E(G) to $$\{1,\ldots ,|E(G)|\}$$ such that any two adjacent vertices have distinct vertex-sums. We prove that every graph with maximum degree at most 4 admits a local antimagic orientation by Alon’s Combinatorial Nullstellensatz.

On the additive chromatic number of several families of graphs

The Additive Coloring Problem is a variation of the Coloring Problem where labels of {1,…,k} are assigned to the vertices of a graph G so that the sum of labels over the neighborhood of each vertex is a proper coloring of G. The least value k for which G admits such labeling is called additive chromatic number of G. This problem was first presented by Czerwiński, Grytczuk and Żelazny who also proposed a conjecture that for every graph G, the additive chromatic number never exceeds the classic chromatic number. Up to date, the conjecture has been proved for complete graphs, trees, non-3-colorable planar graphs with girth at least 13 and non-bipartite planar graphs with girth at least 26. In this work, we show that the conjecture holds for split graphs. We also present exact formulas for computing the additive chromatic number for some subfamilies of split graphs (complete split, headless spiders and complete sun), regular bipartite, complete multipartite, fan, windmill, circuit, wheel, cycle sun and wheel sun.

Radio Quotient Square Sum Labeling of a Graph

Radio Quotient Square Sum Labeling of a Graph

Ulam stability for finite variable quadratic functional equation in Banach algebra

Here, we define a cube sum labeling and cube sum graph. Let G be a (p, q) graph. G is said to be a cube sum graph if there exist a bijection f : V (G) → {0, 1, . . . , p − 1} such that the induced function f ∗ : E(G) → N given by are all distinct. In this paper, we developed the concept of cube sum labeling of some family of graphs like paths, cycle, stars, wheel graph, fan graphs are discussed in this paper.

Cube sum labeling of graphs

Here, we define a cube sum labeling and cube sum graph. Let G be a (p, q) graph. G is said to be a cube sum graph if there exist a bijection f : V (G) → {0, 1, . . . , p − 1} such that the induced function f ∗ : E(G) → N given by are all distinct. In this paper, we developed the concept of cube sum labeling of some family of graphs like paths, cycle, stars, wheel graph, fan graphs are discussed in this paper.

On The Local Edge Antimagic Coloring of Corona Product of Path and Cycle

Let be a nontrivial and connected graph of vertex set and edge set . A bijection is called a local edge antimagic labeling if for any two adjacent edges and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color . The color of each an edge e = uv is assigned bywhich is defined by the sum of label both and vertices and . The local edge antimagic chromatic number, denoted by is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.Keywords: Local antimagic; edge coloring; corona product; path; cycle.

Seventh and Eighth Order Triangular Sum Labeling of Graphs

Seventh and Eighth Order Triangular Sum Labeling of Graphs

On H-Supermagic Labelings of m-Shadow of Paths and Cycles

A simple graph G=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,…,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,…,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is called super if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,…,Gm, and then joining every vertex u in Gi, i∈{1,2,…,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles.

Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs

We investigate the group irregularity strength, sg(G), of a graph, i.e., the least integer k such that taking any Abelian group G of order k, there exists a function f:E(G)→G so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on sg(G) for a general graph G was exponential in n−c, where n is the order of G and c denotes the number of its components. In this note we prove that sg(G) is linear in n, namely not greater than 2n. In fact, we prove a stronger result, as we additionally forbid the identity element of a group to be an edge label or the sum of labels around a vertex. We consider also locally irregular labelings where we require only sums of adjacent vertices to be distinct. For the corresponding graph invariant we prove the general upper bound: Δ(G)+col(G)−1 (where col(G) is the coloring number of G) in the case when we do not use the identity element as an edge label, and a slightly worse one if we additionally forbid it as the sum of labels around a vertex. In the both cases we also provide a sharp upper bound for trees and a constant upper bound for the family of planar graphs.

Minimal sum labeling of graphs

A graph G is called a sum graph if there is a sum labeling of G, i.e., an injective function ℓ:V(G)→N such that for every u,v∈V(G) it holds that uv∈E(G) if and only if there exists a vertex w∈V(G) such that ℓ(u)+ℓ(v)=ℓ(w). We say that sum labeling ℓ is minimal if there is a vertex u∈V(G) such that ℓ(u)=1. In this paper, we show that if we relax the conditions (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth.

Odd-even sum labeling in the context of duplication of graph elements

In this paper, odd-even sum labeling of the graphs obtained by duplication of graph elements of stars and paths are studied

Generalized Polygonal Sum Labeling of Graphs

Generalized Polygonal Sum Labeling of Graphs

On the local edge antimagicness of m-splitting graphs

Let G be a connected and simple graph. A split graph is a graph derived by adding new vertex v′ in every vertex v′ such that v′ adjacent to v in graph G. An m-splitting graph is a graph which has m v′-vertices, denoted by mSpl(G). A local edge antimagic coloring in G = (V, E) graph is a bijection in which for any two adjacent edges e1 and e2 satisfies , where . The color of any edge e = uv are assigned by w(e) which is defined by sum of label both end vertices f(u) and f(v). The chromatic number of local edge antimagic labeling γlea(G) is the minimal number of color of edge in G graph which has local antimagic coloring. We present the exact value of chromatic number γlea of m-splitting graph and some special graphs.

Handicap Labelings of 4-Regular Graphs

Let G be a simple graph, let f : V(G)&#8594{1,2,...,|V(G)|} be a bijective mapping. The weight of v &#8712 V(G) is the sum of labels of all vertices adjacent to v . We say that f is a distance magic labeling of G if the weight of every vertex is the same constant k and we say that f is a handicap magic labeling of G if the weight of every vertex v is l + f(v) for some constant l. Graphs that allow such labelings are called distance magic or handicap, respectively. Distance magic and handicap labelings of regular graphs are used for scheduling incomplete tournaments. While distance magic labelings correspond to so called equalized tournaments, handicap labelings can be used to schedule incomplete tournaments that are more challenging to stronger teams or players, hence they increase competition and yield attractive schemes in which every games counts. We summarize known results on distance magic and handicap labelings and construct a new infinite class of 4-regular handicap graphs.

Characterisation of Graphs with Exclusive Sum Labelling

A sum graph G is a graph with a mapping of the vertex set of G onto a set of positive integers S in such a way that two vertices of G are adjacent if and only if the sum of their labels is an element of S. In an exclusive sum graph the integers of S that are the sum of two other integers of S form a set of integers that label a collection of isolated vertices associated with the graph G. A graph bears a k-exclusive sum labelling (abbreviated k-ESL), if the set of isolated vertices is of cardinality k.In this paper, observing that the property of having a k-ESL is hereditary, we provide a characterisation of graphs that have a k-exclusive sum labelling, for any k≥1, in terms of describing a universal graph for the property.

Square sum labeling for some lilly related graphs

Square sum labeling for some lilly related graphs

More Results on Polygonal Sum Labeling of Graphs

Objectives: To explore and identify some new classes of graphs which exhibit polygonal sum labeling. Methods: In this article we use a methodology which fundamentally involves formulation and subsequent mathematical validation. Findings: Here we establish that the graphs – Star ( K 1,n ), Coconut Tree, Bistar (S m,n ), the Graph S m n k Comb, (P n K 1 ) and Subdivision graph S(K 1,n ) admit pentagonal, hexagonal, heptagonal, octagonal, nonagonal and decagonal sum labeling. Applications: One can explore to generalize these results and extend to give n-gonal labeling to some classes graphs. Sum labeling has already been used in the problems involving relational database management and hence one can try out to use polygonal sum labeling as well in these problems.

Some results on super pair sum graphs

Let G be a graph with P vertices and q edges. The graph G is said to be a super pair sum labeling if there exists a bijection f from when p + q is odd and from when p + q is even such that f(uv) = f(u) + f(v). A graph that admits a super pair sum labeling is called a super pair sum graph. In this paper, we prove that the graphs Hn ⊙ mK1, (P2n; Sm), S’(P2n), < Bm,n : Pk > for m ≥ 1, n ≥ 1, k ≡ 2(mod 4), < B(m): Pk > for m ≥ 1, k ≡ 0, 2(mod 4) and 2Bm,n (m ≥ 1, n ≥ 1) are super pair sum graphs.

Lucky Edge Neighborhood Labeling of Graphs

Let $G$ be a simple graph with vertex set $V(G)$ and edge set $E(G)$ respectively. Vertex set $V(G)$ is labeled arbitrary by positive integers and $E(e)$ denote the edge label such that it is the sum of labels of vertices incident with edge e. A lucky edge neighborhood labeling of $G$ is an assignment of positive integers to the vertices of $G $ so that edge neighborhood labelings are distinct for every edge $e$.The least integer for which a graph $G$ has a lucky edge labeling from the set $\{1,2,...,k\}$ is called the lucky neighborhood number and is denoted by $\eta_N(G)$. In this paper, we prove that $P_n$, $C_n$, $T_{m,n}$, $S(P_n^+)$ and $S(C_n^+)$ are lucky edge neighborhood labeled graphs.