On The Balance Index Sets of Graphs (original) (raw)

On Vertex Balance Index Set of Some Graphs

Let Z2 = {0, 1} and G = (V, E) be a graph. A labeling f : V −→ Z2 induces an edge labeling f * : E −→ Z2 defined by f * (uv) = f (u).f (v). For i ∈ Z2, let v f (i) = v(i) = card{v ∈ V : f (v) = i} and e f (i) = e(i) = card{e ∈ E : f * (e) = i}. A labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. The vertex balance index set is defined by {| e f (0) − e f (1) | : f is vertex-friendly}. In this paper we completely determine the vertex balance index set of Kn, Km,n, Cn × P2 and Complete binary tree.

Balanced Labeling and Balance Index Set of One Point Union of Two Complete Graphs

International Journal of Computer Applications, 2012

Let G be a graph with vertex set V (G) and edge set E(G), and consider the set A = {0, 1}. A labeling f : V (G) → A induces a partial edge labeling f * : E(G) → A defined by f * (xy) = f (x), if and only if f (x) = f (y), for each edge xy ∈ E(G). For i ∈ A, let v f (i) = |{v ∈ V (G) : f (v) = i}| and e f * (i) = |e ∈ E(G) : f * (e) = i|. A labeling f of a graph G is said to be friendly if |v f (0) − v f (1)|≤ 1. A friendly labeling is called balanced if |e f * (0) − e f * (1)|≤ 1. The balance index set of the graph G, Bl(G), is defined as {|e f * (0) − e f * (1)|: the vertex labeling f is friendly}. We provide balanced labeling and balance index set of one point union of two complete graphs.

On the edge-balance index sets of (p, p+ 1)-graphs

2012

Let G be a simple graph with vertex set V (G) and edge set E(G), and let Z 2 = {0, 1}. Any edge labeling f induces a partial vertex labeling f + : V (G) → Z 2 assigning 0 or 1 to f + (v), v being an element of V (G), depending on whether there are more 0-edges or 1edges incident with v, and no label is given to f + (v) otherwise. For each i ∈ Z 2 , let v f (i) = |{v ∈ V (G) : f + (v) = i}| and let e f (i) = |{e ∈ E(G) : f (e) = i}|. An edge-labeling f of G is said to be edge friendly if {|e f (0) − e f (1)| ≤ 1. The edge-balance index set of the graph G is defined as EBI(G) = {|v f (0) − v f (1)| : f is edge-friendly.}. In this paper, we investigate and present results concerning the edge-balance index sets of (p, p + 1)-graphs.

On Balancedness of Some Graph Constructions

Let G be a graph with vertex set V(G) and edge set E(G), and let A = {O, I}. A labeling f: V(G) ~ A induces a partial edge labeling f* : E(G) ~ A defined by f*(xy) = f(x), ifand only if f(x) = fey), for each edge xy E E(G). For i E A, let vrt:i) = card{v E V(G): ftv) = i} and ep(i) = card{e E E(G): f*(e) i}. A labeling f ofa graph G is said to be friendly if I vrt:O) - vrt: I) 1 s: I. If 1 ep(O) ­ ef*( 1) 1 s: 1 then G is said to be balanced. Balancedness of the Cartesian product and composition ofgraphs is studied in (19). We provide some new families of balanced graphs using other constructions.

Balance Index Set of Caterpillar and Lobster Graphs

2016

Let G be a graph with vertex set V (G) and edge set E(G). Consider the set A = {0, 1}. A labeling f : V (G) → A induces a partial edge labeling f * : E(G) → A defined by f * (xy) = f (x), if and only if f (x) = f (y) for each edge xy ∈ E(G). For i ∈ A, let v f (i) = |{v ∈ V (G) : f (v) = i}| and e f * (i) = |{e ∈ E(G) : f * (e) = i}|. A labeling f of a graph G is said to be friendly if |v f (0) − v f (1)| ≤ 1. A friendly labeling is balanced if |e f * (0) − e f * (1)| ≤ 1. The balance index set of the graph G, BI(G), is defined as {|e f * (0) − e f * (1)| : the vertex labeling f is friendly}. In this paper, we obtain the balance index set of caterpillar graphs and lobster graphs.

On Balance and Groups of Labelings on Graphs

2013

Abstract: We discuss functions from the edges of an undirected graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced. By a cycle we mean a closed path with no repeating edges. The set of all the balanced edge functions is a subgroup of the free Abelian group of all the edge functions. We describe the structure of this subgroup and provide an efficient algorithm to compute its order.

Friendly Index Sets and Friendly Index Numbers of Some Graphs

2014

Let G be a graph with vertex set V (G) and edge set E(G). Consider the set A = {0, 1}. A labeling f : V (G) → A, induces a partial edge labeling f * : E(G) → A, defined by f * (xy) = f (x) if and only if f (x) = f (y) for each edge xy ∈ E(G). For i ∈ A, let v f (i) = |{v ∈ V (G) : f (v) = i}| and we denote e Bf * (i) = |{e ∈ E(G) : f * (e) = i}|. In this paper we define friendly index number(FIN) and full friendly index number(FFIN) of graph G as the cardinality of the distinct elements of friendly index set and full friendly index set respectively and obtaining these numbers along with their sets of some families graphs.

On edge-balance index sets of regular graphs

In a bi-racial country, it is desired that the num-bers of government ministers from the two races should differ by at most one for fairness. More-over, the numbers of pairs of ministries which in-teract directly with each other and both headed by ministers of one race should differ by at most one from that of the other race. One can naturally model the above social phenomenon by a graph la-beling which is called edge-balanced, which assigns edges approximately half 0's, and the other half 1's. And then the labeling requires that the in-duced vertex labels are also 0's over approximately one half vertices, and 1's over the other one half vertices. We generalize the concept of edge-balanced labeling to the concept of edge-balance index sets of graphs. In this article, properties regarding the edge-balance index sets of regular graphs are in-vestigated. In particular, we completely determine the edge-balance index sets of all 2-regular graphs, Möbius ladders (as exampl...

On the relations between certain graph labelings

Discrete Mathematics, 1994

This paper deals with certain edge labelings of graphs. After having introduced the concepts of a weak antimagic graph and an Egyptian magic graph, the authors showed that every connected graph of order 23 is weakly antimagic and proved some interesting relations between the different edge labelings. * Corresponding author 0012-365X/94/$07.00 0 1994-Elsevier Science B.V. All rights reserved SSDl 0012-365X(93)E0054-8

On the Sumset Labeling of Graphs

For a positive integer nnn, let mZ\mZmZ be the set of all non-negative integers modulo nnn and sP(mZ)\sP(\mZ)sP(mZ) be its power set. A sumset valuation or a sumset labeling of a given graph GGG is an injective function f:V(G)tosP(mZ)f:V(G) \to \sP(\mZ)f:V(G)tosP(mZ) such that the induced function f+:E(G)tosP(mZ)f^+:E(G) \to \sP(\mZ)f+:E(G)tosP(mZ) defined by f+(uv)=f(u)+f(v)f^+ (uv) = f(u)+ f(v)f+(uv)=f(u)+f(v). A sumset indexer of a graph GGG is an injective sumset valued function f:V(G)tosP(mZ)f:V(G) \to \sP(\mZ)f:V(G)tosP(mZ) such that the induced function f+:E(G)tosP(mZ)f^+:E(G) \to \sP(\mZ)f+:E(G)tosP(mZ) is also injective. In this paper, some properties and characteristics of this type of sumset labeling of graphs are being studied.