Vertex labeling

Let G = (V, E) be a graph with vertices and edges. A graph is an analytic odd mean if there exist an injective function f ∶ V → {0, 1, 3, 5. .. , 2q − 1} with an induce edge labelling * : E → Z such that for each edge with f(u) < (), * () = { ⌈ () 2 − (() + 1) 2 2 ⌉ () ≠ 0 ⌈ () 2 2 ⌉ () = 0 } is injective. We say that is an analytic odd mean labelingof. In this paper we prove that the subdivision and super subdivision of cycle , star 1, , comb ⊙ 1 , path on the comb and-super subdivision of path and cycle are analytic odd mean graphs.

An L(h, k)-labeling of a graph G is an integer labeling of vertices of G, such that adjacent vertices have labels which differ by at least h, and vertices at distance two have labels which differ by at least k. The span of an L(h, k)-labeling is the difference between the largest and the smallest label. We investigate L(h, k)-labelings of trees of maximum degree ∆, seeking those with small span. Given ∆, h and k, span λ is optimal for the class of trees of maximum degree ∆, if λ is the smallest integer such that every tree of maximum degree ∆ has an L(h, k)-labeling with span at most λ. For all parameters ∆, h, k, such that h < k, we construct L(h, k)-labelings with optimal span. We also establish optimal span of L(h, k)-labelings for stars of arbitrary degree and all values of h and k.

The study of set of objects with imprecise knowledge and vague information is known as rough set theory. The diagrammatic representation of this type of information may be handled through graphs for better decision making. Tong He and K. Shi introduced the constructional processes of rough graph in 2006 followed by the notion of edge rough graph. They constructed rough graph through set approximations called upper and lower approximations. He et al developed the concept of weighted rough graph with weighted attributes. Labelling is the process of making the graph into a more sensible way. In this process, integers are assigned for vertices of a graph so that we will be getting distinct weights for edges. Weight of an edge brings the degree of relationship between vertices. In this paper we have considered the rough graph constructed through rough membership values and as well as envisaged a novel type of labeling called Even vertex ζ-graceful labeling as weight value for edges. In case of rough graph, weight of an edge will identify the consistent attribute even though the information system is imprecise. We have investigated this labeling for some special graphs like rough path graph, rough cycle graph, rough comb graph, rough ladder graph and rough star graph etc. This Even vertex ζ-graceful labeling will be useful in feature extraction process and it leads to graph mining.

Let G=(V,E) be a simple graph. A vertex labeling f:V→{0,1} induces a partial edge labeling f * :E→{0,1} defined by f * (uv)=f(u) if and only if f(u)=f(v). For i=0,1, let v f (i)=|{v∈V:f(v)=i}|, and e f (i)=|{e∈E:f * (e)=i}|. A graph G is uniformly balanced if |e f (0)-e f (1)|≤1 for any vertex labeling f that satisfies |v f (0)-v f (1)|≤1. In this paper, we characterize uniformly balanced graphs, and present several ways to construct uniformly balanced graphs.

Abstract: Graham and Sloane [7] introduced the harmonious graphs and Singh & Varkey [8] introduced the odd sequential graphs. Gayathri & Hemalatha [2] introduced even sequential harmonious labeling of graphs. We studied even sequential harmonious labeling of trees in [3]. In [4], we have extended this notion to k-even sequential harmonious labeling graphs. It is further studied in [5]. k-even sequential harmonious labeling of some cycle related graphs are studied in [6 ]. Also, we have introduced k-odd sequential harmonious labeling of graphs in [5]. In this paper, we investigate k-odd sequential harmonious labeling of some graphs.

Let G be a graph with vertex set V and edge set E, and let A be an abelian group. A labeling f : V → A induces an edge labeling f * : E → A defined by f * (x y) = f (x) + f (y). For i ∈ A, let v f (i) = card{v ∈ V : f (v) = i} and e f (i) = card{e ∈ E : f * (e) = i}. A labeling f is said to be A-friendly if |v f (i) − v f (j)| ≤ 1 for all (i, j) ∈ A × A, and A-cordial if we also have |e f (i) − e f (j)| ≤ 1 for all (i, j) ∈ A × A. When A = Z 2 , the friendly index set of the graph G is defined as {|e f (1) − e f (0)| : the vertex labeling f is Z 2-friendly}. In this paper we completely determine the friendly index sets of 2-regular graphs. In particular, we show that a 2-regular graph of order n is cordial if and only if n ≡ 2 (mod 4).

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 1-edges 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 wheels.

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.

In 1991, Gnanajothi [4] proved that the path graph n P with n vertex and 1 n − edge is odd graceful, and the cycle graph m C with m vertex and m edges is odd graceful if and only if m even, she proved the cycle graph is not graceful if m odd. In this paper, firstly, we studied the graph m n C P ∪ when 4, 6, 8,10 m = and then we proved that the graph m n C P ∪ is odd graceful if m is even. Finally, we described an algorithm to label the vertices and the edges of the vertex set

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 1-edges 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 wheels.

In general, the logarithmic mean of two positive integers need not be an integer. Hence, the logarithmic mean is to be an integer; we use either flooring or ceiling function. The logarithmic mean labeling of graphs have been defined in which the edge labels may be assigned by either flooring function or ceiling function. In this, we establish the logarithmic mean labeling on graphs by considering the edge labels obtained only from the flooring function. A logarithmic mean labeling of a graph G with q edges is an injective function from the vertex set of G to 1, 2, 3,..., q+1 such that the edge labels obtained from the flooring function of logarithmic mean of the vertex labels of the end vertices of each edge are all distinct, and the set of edge labels is 1, 2, 3,..., q. A graph is said to be a logarithmic mean graph if it admits a logarithmic mean labeling. In this paper, we study the logarithmic meanness of some ladder related graphs.

In a study of traffic, the labelling problems in graph theory can be used by considering the crowd at every junction as the weights of a vertex and expected average traffic in each street as the weight of the corresponding edge. If we assume the expected traffic at each street as the arithmetic mean of the weight of the end vertices, that causes mean labelling of the graph. When we consider a geometric mean instead of arithmetic mean in a large population of a city, the rate of growth of traffic in each street will be more accurate. The geometric mean labelling of graphs have been defined in which the edge labels may be assigned by either flooring function or ceiling function. In this, the readers will get some confusion in finding the edge labels which edge is assigned by flooring function and which edge is assigned by ceiling function. To avoid this confusion, we establish the F-geometric mean labelling on graphs by considering the edge labels obtained only from the flooring funct...

The objective of this paper is to present a new class of odd graceful graphs. In particular, we show that the linear cyclic snakes (1, k) C 4-snake and (2, k) C 4-snake are odd graceful. We prove that the linear cyclic snakes (1, k) C 6-snake and (2, k) C 6-snake are odd graceful. We also prove that the linear cyclic snakes (1, k) C 8-snake and (2, k) C 8-snake are odd graceful. We generalize the above results "the linear cyclic snakes (m, k) C 4-snake, (m, k) C 6-snake and (m, k) C 8-snake are odd graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) C n-snakes are odd graceful if n is even)".

Let G be a graph with vertex set V(G) and edge set E(G), and f be a 0−1 labeling of E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling fedge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident

The objective of this paper is to present a new class of odd graceful graphs. In particular, we show that the linear cyclic snakes (1, k) C 4-snake and (2, k) C 4-snake are odd graceful. We prove that the linear cyclic snakes (1, k) C 6-snake and (2, k) C 6-snake are odd graceful. We also prove that the linear cyclic snakes (1, k) C 8-snake and (2, k) C 8-snake are odd graceful. We generalize the above results "the linear cyclic snakes (m, k) C 4-snake, (m, k) C 6-snake and (m, k) C 8-snake are odd graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) C n-snakes are odd graceful if n is even)".

The aim of this paper is to present a new class of graceful graphs. In particular, we show that linear cyclic snakes (1, k) C 4-snake and (2, k) C 4-snake are graceful. We also prove that linear cyclic snakes (1, k) C 8-snake and (2, k) C 8-snake are graceful. We generalize the above results " the linear cyclic snakes (m, k) C 4-snake and (m, k) C 8-snake are graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) C n-snakes are graceful if n  0 (mod 4) or n  3 (mod 4)".

The objective of this paper is to present a new class of odd
graceful graphs. In particular, we show that the linear cyclic
snakes (1, k) C4- snake and (2, k) C4- snake are odd graceful.
We prove that the linear cyclic snakes (1, k) C6- snake and (2,
k) C6- snake are odd graceful. We also prove that the linear
cyclic snakes (1, k) C8- snake and (2, k) C8- snake are odd
graceful. We generalize the above results "the linear cyclic
snakes (m, k) C4- snake, (m, k) C6-snake and (m, k) C8-snake
are odd graceful ". Finally, we introduce a new conjecture"
All the linear cyclic snakes (m, k) Cn-snakes are odd graceful
if n is even)".

The aim of this paper is to present a new class of graceful graphs. In particular, we show that linear cyclic snakes (1, k) C4- snake and (2, k) C4- snake are graceful. We also prove that linear cyclic snakes (1, k) C8- snake and (2, k) C8- snake are graceful. We generalize the above results " the linear cyclic snakes (m, k) C4- snake and (m, k) C8- snake are graceful ". Finally, we introduce a new conjecture" All the linear cyclic snakes (m, k) Cn-snakes are graceful if n  0 (mod 4 ) or n  3 (mod 4 )"