Sum Of Labels Research Articles

Odd-Even Sum Labeling of some Graphs

A $(p,q)$ graph $G=(V,E)$ is said to be an odd-even sum graph if there exists an injective function $f:V(G)\rightarrow\lbrace\pm 1,\pm 3 \pm 5, ...,\pm (2p-1)\rbrace$ such that the induced mapping $f^{*}:E(G)\rightarrow\lbrace 2,4,6, ...,2q\rbrace$ defined by $f^{*}(uv)=f(u)+f(v)~\forall~uv\in E(G)$ is bijective. The function $f$ is called an odd-even sum labeling of $G$. In this paper we study odd-even sum labeling of path $P_{n}(n\geq2)$, star $K_{1,n}(n\geq 1)$, bistar $B_{m,n}$,$S(K_{1,n})$, $B(m,n,k)$ and some standard graphs.

О квадратной суммарной разметке некоторых графов

О квадратной суммарной разметке некоторых графов

English

English

Edge pair sum labeling of some cartesian product of graphs

Let [Formula: see text] be a [Formula: see text] graph. An injective map [Formula: see text] is said to be an edge pair sum labeling if the induced vertex function [Formula: see text] defined by [Formula: see text] is one-to-one where [Formula: see text] denotes the set of edges in [Formula: see text] that are incident with the vertex [Formula: see text] and [Formula: see text] is either of the form [Formula: see text] or [Formula: see text] accordingly as [Formula: see text] is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper, we prove that the graphs [Formula: see text], [Formula: see text], book graph and [Formula: see text] admit edge pair sum labeling.

ADCSS-Labeling of 2-Tuple Graphs of Some Graphs

A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. In this paper, we introduce the new concept, an absolute difference of cubic and square sum labeling of a graph. The graph for which every edge label is the absolute difference of the sum of the cubes of the end vertices and the sum of the squares of the end vertices. It is also observed that the weights of the edges are found to be multiples of 2. Here we characterize 2-tuple graphs of middle graph of paths and cycles, crown graph ,star graphs , the triangular snake graph, quadrilateral snake graph for adcss labeling.

Edge pair sum labeling of butterfly graph with shell order

The concept of an edge pair sum labeling was introduced in [3]. Let $G(p, q)$ be a graph. An injective map $f: E(G) \rightarrow\{ \pm 1, \pm 2, \cdots, \pm q\}$ is said to be an edge pair sum labeling if the induced vertex function $f^*: V(G) \rightarrow Z-\{0\}$ defined by $f^*(v)=\sum_{e \epsilon E_v} f(e)$ is one- one where $E_v$ denotes the set of edges in $G$ that are incident with a vertex $v$ and $f^*(V(G))$ is either of the form $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p}{2}}\right\}$ or $\left\{ \pm k_1, \pm k_2, \cdots, \pm k_{\frac{p-1}{2}}\right\}$ $\cup\left\{ \pm k_{\frac{p+1}{2}}\right\}$ according as $p$ is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the shell graph and butterfly graph with shell order are edge pair sum graphs.

Some new formulae on Δ–optimum exclusive sum labeling of certain trees

All sum graphs are disconnected. The notions of sum labeling and sum graphs were introduced by Harary [1] and later extended to include all integers [2]. A mapping L is called a sum labeling a graph H(V (H), E(H)) if it is an injection from V(H) to a set of positive integers such that xy ∈E(H) if and only if there exists a vertex w ∈V(H) such that L(w)=L(x) + L(y). In this case w is called a working vertex. We call L as an exclusive sum labeling of a graph G if it is sum labeling of for some non negative integer r and G contains no working vertex. In general, a graph G will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called exclusive sum number of G, denoted by ∈(G). An exclusive sum labeling of a graph G is said to be optimum if it labels G exclusively by using ∈(G) isolated vertices. In case ∈(G)=Δ(G), where D(G) denotes the maximum degree of vertices in G, the labeling is called Δ- optimum exclusive sum labeling. In this paper we developed some new formulae to find Δ- optimum exclusive sum labeling of certain trees.

Bimagic Labeling for Certain Families of Strong Face Graphs

Let G = (V, E, F) be a simple, finite, connected plane graph with V(G) as a vertex set, E(G) an edge set and F(G) set of faces, a labeling of type (1, 1, 0) is a bijection from the vertex set and edge set of a graph G to the set {1, 2, …, |V(G)|+|E(G)|}. This labeling is called super if the set of vertices received the smallest number. The weight of a face w(f) under this labeling is sum of labels of the vertices and edges surrounding this face.

Odd Sum Labeling of Graphs Obtained by Duplicating Any Edge of Some Graphs

An injective function $f:V(G)\rightarrow \{0,1,2,\dots,q\}$ is an odd sum labeling if the induced edge labeling $f^*$ defined by $f^*(uv)=f(u)+f(v),$ for all $uv\in E(G),$ is bijective and $f^*(E(G))=\{1,3,5,\dots,2q-1\}.$ A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.

Some Edge Pair Sum Graphs

An injective map f : E(G)→{± 1,±2, … ,±q} is said to be an edge pair sum labeling of a graph G(p, q) if the induced vertex function f* : V(G)→Z − {0} defined by f* (v)=∑e∈Evf(e) is one – one, where Ev denotes the set of edges in G that are incident with a vertex v and f*(V(G)) is either of the form or according as p is even or odd. A graph which admits edge pair sum labeling is called an edge pair sum graph. In this paper we prove that the one point union of cycles, the perfect binary tree, shadow graph, total graph and Pn2 admit edge pair sum labeling.

Some results on edge pair sum labeling

An injective map f:E(G)→{±1,±2,⋯,±q} is said to be an edge pair sum labeling if the induced vertex function f⁎:V(G)→Z−{0} defined by f⁎(v)=∑eϵEvf(e) is one-one where Ev denotes the set of edges in G that are incident with a vertex v and f⁎(V(G)) is either of the form {±k1,±k2,⋯,±kp2} or {±k1,±k2,⋯,±kp−12}⋃{kp+12} according as p is even or odd. A graph with an edge pair sum labeling is called an edge pair sum graph. In this paper we prove that triangular snake, bistars, Cn⋃Cn and K1,n⋃K1,m are edge pair sum graphs.

Vertex Antimagic Total Labeling of Digraphs

In this paper we investigate the properties of (a, d)-vertex antimagic total labeling of a digraph D = (V, A). In this labeling, we assign to the vertices and arcs the consecutive integers from 1 to |V|+|A| and calculate the sum of labels at each vertex, i.e., the vertex label added to the labels on its out arcs. These sums form an arithmetical progression with initial term a and common difference d. We show the existence and non-existence of (a, d)-vertex antimagic total labeling for several class of digraphs, and show how to construct labelings for generalized de Bruijn digraphs. We conclude this paper with an open problem suitable for further research.

Antimagic Labelings of Join Graphs

An antimagic labeling of a graph with q edges is a bijection from the set of edges of the graph to the set of positive integers \({\{1, 2,\dots,q\}}\) such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. The join graph G + H of the graphs G and H is the graph with \({V(G + H) = V(G) \cup V(H)}\) and \({E(G + H) = E(G) \cup E(H) \cup \{uv : u \in V(G) {\rm and} v \in V(H)\}}\). The complete bipartite graph Km,n is an example of join graphs and we give an antimagic labeling for \({K_{m,n}, n \geq 2m + 1}\). In this paper we also provide constructions of antimagic labelings of some complete multipartite graphs.

Square Sum Labeling of Class of Planar Graphs

A (p, q) graph G is said to be square sum, if there exists a bijection f : V(G) → {0,1, 2,...,p — 1} such that the induced function f * : E(G)→ N defined by f * (uv) = (f (u))2 + (f (v))2, ∀ uv ∈ E(G) is injective. In this paper we proved that the planar graphs Plm,n,TBL(n,α,k,β) and higher order level joined planar grid admits square sum labeling. Also the square sum properties of several classes of graphs with many odd cycles are studied.

On Face Magic Labeling of Duplication Graphs

A labeling of a graph G of type assigns labels from the set to the vertices, edges and faces of G such that each vertex receives a label(s), each edge receives b label(s) and each face receives c label(s) and each number is used exactly once as a label. We restrict a, b and c to be not greater than one. The weight of face w(f) under a labeling is the sum of labels of the face itself together with labels of vertices and edges surrounding that face. A labeling is said to be magic, if for every positive integer s, all s-sided faces have the same weight. In this paper, we examine whether vertex and edge duplication on some families of graphs admit face magic labeling of types (1, 0, 1), (1, 1, 0), (0, 1, 1) and (1, 1, 1).

Few More Results on Sum Labeling of Split Graphs

Few More Results on Sum Labeling of Split Graphs

Regular Graphs of Odd Degree Are Antimagic

AbstractAn antimagic labeling of a graph G with m edges is a bijection from to such that for all vertices u and v, the sum of labels on edges incident to u differs from that for edges incident to v. Hartsfield and Ringel conjectured that every connected graph other than the single edge K2 has an antimagic labeling. We prove this conjecture for regular graphs of odd degree.

Topological additive numbering of directed acyclic graphs

We propose to study a problem that arises naturally from both Topological Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as Lucky Labeling). Let D be a digraph and f a labeling of its vertices with positive integers; denote by S(v) the sum of labels over all neighbors of each vertex v. The labeling f is called topological additive numbering if S(u)<S(v) for each arc (u,v) of the digraph. The problem asks to find the minimum number k for which D has a topological additive numbering with labels belonging to {1,…,k}, denoted by ηt(D).We characterize when a digraph has topological additive numberings, give a lower bound for ηt(D) and provide an integer programming formulation for our problem. We also present some families for which ηt(D) can be computed in polynomial time. Finally, we prove that this problem is NP-Hard even when its input is restricted to planar bipartite digraphs.

The 1,2,3-Conjecture and 1,2-Conjecture for sparse graphs

The 1, 2, 3-Conjecture states that the edges of a graph without isolated edges can be labeled from {1, 2, 3} so that the sums of labels at adjacent vertices are distinct. The 1, 2-Conjecture states that if vertices also receive labels and the vertex label is added to the sum of its incident edges, then adjacent vertices can be distinguished using only {1, 2}. We show that various configurations cannot occur in minimal counterexamples to these conjectures. Discharging then confirms the conjectures for graphs with maximum average degree less than 8/3. The conjectures are already confirmed for larger families, but the structure theorems and reducibility results are of independent interest.

Odd sum labeling of some subdivision graphs

An injective function ƒ : V(G) →{0,1, 2,..., q} is an odd sum labeling if the induced edge labeling ƒ* defined by ƒ*(uv) = ƒ (u) + ƒ(v), for all uvє E(G), is bijective and ƒ*(E(G)) = {1, 3, 5,..., 2q - 1}. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper, we have studied the odd sum property of the subdivision of the triangular snake, quadrilateral snake, slanting ladder, Cp ○ K1, H ○ K1, Cm@Cn, the grid graph Pm x Pn, duplication of a vertex of a path and duplication of a vertex of a cycle.