Edge Uv Research Articles

Neighbor sum distinguishing total choosability of planar graphs without adjacent special 5-cycles

A total k-coloring of a graph G is a mapping ϕ: V(G)∪E(G)→{1,2,…,k} such that no two adjacent or incident elements in V(G)∪E(G) receive the same color. In a total k-coloring of G, let w(v) denote the total sum of colors of the edges incident with v and the color of v. If for each edge uv∈E(G), w(u)≠w(v), then we call such a total k-coloring neighbor sum distinguishing. Let χ∑′′(G) denote the smallest number k in such a coloring of G. Pilśniak and Woźniak posed the conjecture that χ∑′′(G)≤Δ(G)+3 for any simple graph G with maximum degree Δ(G). In this paper, we focus on the list version of neighbor sum distinguishing total coloring. Let Lz (z∈V(G)∪E(G)) be a set of lists of integer numbers, each of size k. The smallest k for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from Lz for each z∈V(G)∪E(G) is called the neighbor sum distinguishing total choosability of G, and denoted by ch∑′′(G). We prove that ch∑′′(G)≤Δ(G)+3 for planar graphs without adjacent special 5-cycles with Δ(G)≥8.

Mean Square Cordial Labeling on Star Related Graphs

A Mean Square Cordial Labeling (MSCL) of a Graph G(V, E) with p vertices and q edges is a bijection from V to {0, 1} such that each edge uv is assigned the label where ⌈x⌉ (ceil( x)) is the least integer greater than or equal to x with the condition that the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled 0 and the number of edges labeled with 1 differ by at most 1. In this paper it is analyzed that some star related graphs like Bistar graph B(k, k), Subdivision of a star graph S(Sk), Coconut tree CT(k, k), Merge graph Ck*Sk,, Graph PmΘ K1, n and Banana tree BT(k, k) admits mean square cordial labeling.

Some reduction operations to pairwise compatibility graphs

A graph G=(V,E) with a vertex set V and an edge set E is called a pairwise compatibility graph (PCG, for short) if there are a tree T whose leaf set is V, a non-negative edge weight w in T, and two non-negative reals dmin≤dmax such that G has an edge uv∈E if and only if the sum of the edge weights in the unique path between u and v in the weighted tree (T,w) is in the interval [dmin,dmax]. PCG is a relatively new graph class motivated from bioinformatics. In this paper, we give some necessary conditions and sufficient conditions for a graph to be in PCG based on cut-vertices and twins, which provide reductions among PCGs. Our results imply that complete k-partite graphs, cacti, and some other graph classes are subsets of PCG.

Extended results on sum divisor cordial labeling

A sum divisor cordial labeling of a graph G with vertex set V (G) is a bijection f : V (G) → {1, 2, ..., |V (G)|} such that an edge uv assigned the label 1 if 2 divides f(u)+f(v) and 0 otherwise. Further the number of edges labeled with 0 and the the number of edges labeled with 1 differ by atmost 1. A graph with sum divisor cordial labeling is called a sum divisor cordial graph. In this paper we prove that the graphs Pn + Pn (n is odd), Pn@K1,m, Cn@K1,m (n is odd), Wn ∗ K1,m (n is even), , are sum divisor cordial graphs.

A note on edge-distance-balanced property of the generalized Petersen graphs GP(4t, 2)

A graph G is said to be edge-distance-balanced if for any edge uv of G, the number of edges closer to u than to v is equal to the number of edges closer to v than to u. Let GP(n, k) be a generalized Petersen graph. It is proven that for any integers t ≥ 5, the generalized Petersen graph GP(4t, 2) is not edge-distance-balanced.

Influence of Ni and Sr on the structural, morphological and optical properties of ZnO synthesized by sol gel

ZnO nanoparticles doped and co-doped with Ni and Sr were synthesized by the sol gel method. Rietveld refinement indicated the single phase of ZnO for all the samples and suggests that the variations in structural parameters are dependent on the ions type. Fourier transform infrared studies showed the functional groups and chemical bonding of ZnO and confirmed the replacement of Zn by Ni and Sr ions. The micrographs showed particles with clusters shapes and different sizes. Photoluminescence measurements show a shift in near band edge UV emission, from 380 to 384 nm for doped and co-doped samples. Additionally, the Zn substitution by Ni and Sr ions minimizes the intrinsic defects in the ZnO structure. The optical study showed that the replacement of Zn by Ni decreases the band gap, whereas the substitution by Sr increases it. For co-doped samples, this value depends on the dopant concentration and the ions type. This work contributes to the understanding of optical properties for the ZnO nanoparticles co-doping with Ni and Sr, which may be important for future applications.

A Special Research on Homo Cordial Labeling of Spider Graph

Let G= (V, E) be a graph with p vertices and q edges. A Homo Cordial Labeling of a graph G with vertex set V is a bijection from V to {0, 1} such that each edge uv is assigned the label 1 if f(u)=f(v) or 0 if f u f v ( ) ( )  with the condition that the number of vertices labelled with 0 and the number of vertices labelled with 1 differ by at most 1 and the number of edges labelled with 0 and the number of edges labelled with 1 differ by at most 1. The graph that admits a Homo- Cordial labelling is called Homo Cordial graph. In this paper we prove that the spider graph 1 ,2  m t SP is Homo-Cordial labelling graph and further study on the generalisation of labelling spider graph 1 ,2 

Quotient Square Sum Cordial Labeling

Let G = (V, E) be a simple graph and : V {1, 2, ... | V |} be a bijection, for each edge uv assigned the label 1 if is odd and 0 if is even. is called quotient square sum cordial labeling if | (0) − (1)| ≤ 1, where (0) and (1) denote the number of edges labeled with 0 and labeled with 1 respectively. A graph which admits a quotient square sum cordial labeling is called quotient square sum cordial graph. In this paper path Pn, cycle Cn, star K1,n , friendship graph Fn , bistar Bn,n , C4 ∪ Pn ,Km,2 and Km,2 ∪ Pn are shown to be quotient square sum cordial labeling .

On Edge Irregular Total k-labeling and Total Edge Irregularity Strength of Barbell Graphs

Let G be a connected graph with a non empty vertex set V(G) and edge set E(G). An edge irregular total k-labeling of a graph G is a labeling λ : V(G) ⋃ E(G) → {1, 2, …, k}, so that every two different edges have different weights. The weight of edge uv of G is the sum of the labels vertices u and v and label of the edge uv, which is can be written as wt(uv) = λ (u) + λ (uv) + λ(v). The total edge irregularity strength of G, denoted by tes(G) is the minimum positive integer k for which the graph G has an edge irregular total k-labeling. Barbell graph Bn is obtained by connecting two copies of a complete graph Kn by a bridge. In this research, we determined the total edge irregularity strength of barbell graph Bn for n ≥ 3.

Backbone Coloring of Graphs with Galaxy Backbones

A (proper) k-coloring of a graph G = (V,E) is a function c : V (G) → {1,...,k} such that c(u) ≠ c(v) for every uv ∈ E(G). Given a graph G and a subgraph H of G, a q-backbone k-coloring of (G,H) is a k-coloring c of G such that q ≤ |c(u) − c(v)| for every edge uv ∈ E(H). The q-backbone chromatic number of (G,H), denoted by BBCq(G,H), is the minimum integer k for which there exists a q-backbone k-coloring of (G,H). Similarly, a circular q-backbone k-coloring of (G,H) is a function c: V (G) → {1,...,k} such that, for every edge uv ∈ E(G), we have |c(u)−c(v)| ≥ 1 and, for every edge uv ∈ E(H), we have k−q ≥ |c(u)−c(v)| ≥ q. The circular q-backbone chromatic number of (G,H), denoted by CBCq(G,H), is the smallest integer k such that there exists such coloring c.In this work, we first prove that if G is a 3-chromatic graph and F is a galaxy, then CBCq(G,F) ≤ 2q + 2. Then, we prove that CBC3(G,M) ≤ 7 and CBCq(G,M) ≤ 2q, for every q ≥ 4, whenever M is a matching of a planar graph G. Moreover, we argue that both bounds are tight. Such bounds partially answer open questions in the literature. We also prove that one can compute BBC2(G,M) in polynomial time, whenever G is an outerplanar graph with a matching backbone M. Finally, we show a mistake in a proof that BBC2(G,M) ≤ Δ(G)+1, for any matching M of an arbitrary graph G [Miškuf et al., 2010] and we present how to fix it.

On Edge-magic Labelings of Forests

Given an n-vertex graph G = (V,E) with m edges, a labeling f of V ∪ E that uses all the labels in the set {1,2,...,n + m} is edge-magic if there is an integer k such that f(u) + f(v) + f(uv) = k for every edge uv ∈ E. Furthermore, if the labels in {1,2,...,n} are given to the vertices, then f is called super edge-magic. Kotzig [On magic valuations of trichromatic graphs, Reports of the CRM, 1971] started the investigation of super edge-magic labelings of forests. Following this line of research, we prove that some forests of stars admit a super edge-magic labeling and that some forests of caterpillars admit an edge-magic labeling.

Subtract Divisor Cordial Labeling

A subtract divisor cordial labeling is bijection r: Z (G+ ) → {1,2,…,|V(G+ )|} in such a way that an edge uv give the label 1 if r(u) - r(v) is divisible by 2 otherwise give the label 0, then absolute difference of number of edges having label 1 and 0 is at most 1. A graph which fulfill the condition of subtract divisor cordial labeling is called subtract divisor cordial graph. In given paper, we found ten new graphs satisfying the condition of subtract divisor cordial labeling. AMS Subject classification number: 05C78

Intersection cordiality of union of some standard graphs

An intersection cordial labeling of a graph G with vertex set V is an injection f from V to the power set of {1, 2, … , n} such that if each edge uv is assigned the label 1 if f(u) ∩ f(v) ≠ Ø and 0 otherwise; Then the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. If a graph has an intersection cordial labeling, then it is called intersection cordial graph. In this paper, we are going to check whether the graphs 2kP2n, 2kC2n, 2kK1, 2n–1 and 2kK2, 2n–2 are intersection cordial or not.

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.

On the strength of some trees

Let G be a graph of order p. A numbering f of G is a labeling that assigns distinct elements of the set 1,2,…,p to the vertices of G, where each edge uv of G is labeled fu+fv. The strength strfG of a numbering f:VG→1,2,…,p of G is defined by strfG=maxfu+fvuv∈EG,that is, strfG is the maximum edge label of G, and the strength strG of a graph G itself is strG=minstrfGf is a numbering of G.The strengths strT and strTn,k are determined for caterpillars T and k-level complete n-ary trees Tn,k. The strength strG is also given for graphs G obtained by taking the corona of certain graphs and an arbitrary number of isolated vertices. The work of this paper suggests an open problem on the strength of trees.

On [formula omitted]-labellings of lobsters and trees with a perfect matching

A graceful labelling of a tree T is an injective function f:V(T)→{0,…,|E(T)|} such that {|f(u)−f(v)|:uv∈E(T)}={1,…,|E(T)|}. An α-labelling of a tree T is a graceful labelling f with the additional property that there exists an integer k∈{0,…,|E(T)|} such that, for each edge uv∈E(T), either f(u)≤k<f(v) or f(v)≤k<f(u). In this work, we prove that the following families of trees with maximum degree three have α-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; trees T with a perfect matching M such that the contraction T∕M has a balanced bipartition and an α-labelling; and trees with a perfect matching such that their contree is a caterpillar with a balanced bipartition. These results are a step towards the conjecture posed by Bermond in 1979 that all lobsters have graceful labellings and also reinforce a conjecture posed by Brankovic, Murch, Pond and Rosa in 2005, which says that every tree with maximum degree three and a perfect matching has an α-labelling.

Vertex coloring edge-weighting of coronation by path graphs

In this paper, we study vertex coloring edge of corona graph. A k-edge weigting of graph G is mapping . An edge-weighting w induces a vertex coloring defined by . An edge-weighting w is vertex coloring if for any edge uv. Chang, et.al denoted by µ(G) the minimum k for which G has a vertex-coloring k-edge-weighting. In this paper, we obtain the lower bound of of vertex coloring edge-weighting of and we study the exact value of vertex coloring edge-weighting of path corona several graph.

On the total edge irregularity strength of three towers hanoi graph

Let G(V, E) be a connected, simple, and undirected graph with vertex set V and edge set E. A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, … k}. An edge irregular total k-labeling of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G, denoted by wt(uv), is defined as the sum of the label of u, the label of v, and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. The three towers hanoi graph is the state graph for Tower of Hanoi problems with three towers. In this paper, we investigate the total edge irregularity strength of three towers hanoi graph for n ≥ 1. The result is .

Extremal bicyclic graphs with respect to Mostar index

For an edge uv of a graph G, nu denotes the number of vertices of G closer to u than to v, and similarly nv is the number of vertices closer to v than to u. The Mostar index of a graph G is defined as the sum of absolute differences between nu and nv over all edges uv of G. In the paper we prove a recent conjecture of Došlić et al. (2018) on a characterization of bicyclic graphs with given number of vertices, for which extremal values of Mostar index are attained.

3-difference cordiality of some corona graphs

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map where k is an integer 2 ≤ k ≤ p. For each edge uv, assign the label |f (u) − f (v)|. f is called k-difference cordial labeling of G if |vf (i) − vf (j)| ≤ 1 and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the umber of vertices labelled with x, ef (1) and ef (0) respectively denote the number of edges labelled with 1 and not labelled with 1. A graph with a k-difference cordial labeling is called a k-difference cordial graph. In this paper we investigate 3-difference cordial labeling behavior of Tn ʘK1, Tn ʘ2K1, Tn ʘK2, A(Tn)ʘK1, A(Tn)ʘ 2K1, A(Tn) ʘ K2.