Edge Uv Research Articles

GROUP MEAN CORDIAL LABELING OF SOME QUADRILATERAL SNAKE GRAPHS

Let G be a (p, q) graph and let A be a group. Let f : V (G) −→ A be a map. For each edge uv assign the label [o(f (u))+o(f (v)) / 2]. Here o(f (u)) denotes the order of f (u) as an element of the group A. Let I be the set of all integerslabeled by the edges of G. f is called a group mean cordial labeling if the following conditions hold: (1) For x, y ∈ A, |vf (x) − vf (y)| ≤ 1, where vf (x) is the number of vertices labeled with x. (2) For i, j ∈ I, |ef (i) − ef (j)| ≤ 1, where ef (i) denote the number of edges labeled with i. A graph with a group mean cordial labeling is called a group mean cordial graph. In this paper, we take A as the group of fourth roots of unity and prove that, Quadrilateral Snake, Double Quadrilateral Snake, Alternate Quadrilateral Snake and Alternate Double Quadrilateral Snake are groupmean cordial graphs.

EDGE IRREGULAR REFLEXIVE LABELING OF DUMBBELL GRAPH, CORONA OF OPEN LADDER, AND NULL GRAPH

Graph is a simple, connected, undirected graph with vertex set and edge set . A graph is called to have an edge irregular reflexive -labeling if its vertices can be labeled with even numbers from until and its edges can be labeled with positive integers from to such that the weights for all the edges are different, where . The weight of edge uv in graph with labeling, denoted by , is defined as sum of the edge label and all vertex labels incident to that edge. The reflexive edge strength of a graph , denoted by , is the value of minimum of the largest label. In this paper, edge irregular reflexive -labeling for Dumbbell Graph and corona of open ladder and null graph will be determined. The reflexive edge strength of the Dumbbell Graph with and is for and for The reflexive edge strength of the corona of open ladder and null graph with n ≥ 3 and m ≥ 1 is for and for .

Arithmetic Sequential Graceful Labeling For Arrow Graphs

Let G be a simple, finite, connected, undirected, non-trivial graph with 𝑝 vertices and 𝑞 edges. 𝑉(𝐺) be the vertex set and 𝐸(𝐺) be the edge set of 𝐺. Let 𝑓: 𝑉(𝐺) → {𝑎, 𝑎 + 𝑑, 𝑎 + 2𝑑, 𝑎 + 3𝑑, … , 𝑎 + 2𝑞𝑑} where a ≥ 0 and 𝑑 ≥ 1 is an injective function. If for each edge 𝑢𝑣 ∈ 𝐸(𝐺) , 𝑓 ∗ : 𝐸(𝐺) → {𝑑, 2𝑑, 3𝑑, 4𝑑, … , 𝑞𝑑} defined by 𝑓 ∗ (𝑢𝑣) = |𝑓(𝑢)− 𝑓(𝑣)| is a bijective function then the function 𝑓 is called arithmetic sequential graceful labeling. The graph with arithmetic sequential graceful labeling is called arithmetic sequential graceful graph. In this paper, we prove that one side arrow graphs 𝐴𝑅𝜂 2 , 𝐴𝑅𝜂 3 , 𝐴𝑅𝜂 5 and double-sided arrow graphs 𝐷(𝐴𝑅𝜂 2 ),𝐷(𝐴𝑅𝜂 3 ) are arithmetic sequential graceful graph

The diameter of sum basic equilibria games

We study the sum basic network creation game introduced in 2010 by Alon, Demaine, Hajiaghai and Leighton. In this game, an undirected and unweighted graph G is said to be a sum basic equilibrium if and only if, for every edge uv and any vertex v′ in G, swapping edge uv with edge uv′ does not decrease the total sum of the distances from u to all the other vertices. This concept lies at the heart of the network creation games, where the central problem is to understand the structure of the resulting equilibrium graphs, and in particular, how well they globally minimize the diameter. In this sense, in 2013 Alon et al. showed an upper bound of 2O(log⁡n) on the diameter of sum basic equilibria, and they also proved that if a sum basic equilibrium graph is a tree, then it has diameter at most 2. In this paper, we prove that the upper bound of 2 also holds for bipartite graphs and even for some non-bipartite classes like block graphs and cactus graphs.

Argon plasma-enhanced UV light emission from ZnO submicrowires grown by hydrothermal method

In this work, the optical, morphological, and structural improvements of vertically aligned, hydrothermally grown ZnO submicrowires (SMWs) treated with a 250 W, 250 V RF argon plasma (Ar) during different exposure times were investigated by scanning and transmission electrons microscopy, X-ray diffraction, and micro-Raman and photoluminescence (PL) spectroscopies. In two steps, the SMWs were synthesized in an aqueous medium at low temperatures. The plasma-treated samples showed significantly improved room-temperature PL compared to untreated samples. All the treated samples exhibited a substantial increase of the near band edge UV emission intensity and a decrease of the deep level emission in the visible. The samples treated for 4 minutes presented the best UV/vis intensity ratio of ∼ 1568 (an increase of ⁓174-fold with respect to the untreated sample). The analysis of the UV band in terms of the first and second phonon replica of the excitonic emission indicated that the Ar plasma treatment favors the multiple phonon-exciton coupling, with partial correlation with the observed overall increase of the UV/visible intensity ratio. From the PL measurements at low temperatures, the presence of excitons bound to H donors in the ZnO structure was inferred. The possible reasons for the Ar plasma-induced enhancement of the UV emission from the treated SMWs are discussed in terms of previous work, the observed morphology and changes expected to occur at the SMW surfaces due to Ar ion impacts from the plasma.

On the Extremal Weighted Mostar Index of Bicyclic Graphs

Let G be a simple connected graph with edge set E(G) and vertex set V(G). The weighted Mostar index of a graph G is defined as w+Mo(G)=∑e=uv∈E(G)(dG(u)+dG(v))|nu(e)−nv(e)|, where nu(e) denotes the number of vertices closer to u than to v for an edge uv in G. In this paper, we obtain the upper bound and lower bound of the weighted Mostar index among all bicyclic graphs and characterize the corresponding extremal graphs.

On edge irregularity strength of cycle-star graphs

For a simple graph G, a vertex labeling ϕ : V (G) → {1, 2, . . . , k} is called k-labeling. The weight of an edge uv in G, written wϕ(uv), is the sum of the labels of end vertices u and v, i.e., wϕ(uv) = ϕ(u) + ϕ(v). A vertex k-labeling is defined to be an edge irregular k-labeling of the graph G if for every two distinct edges u and v, wϕ(u) ̸= wϕ(v). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, written es(G). In this paper, we study the edge irregular k-labeling for cycle-star graph CSk,n−k and determine the exact value for cycle-star graph for 3 ≤ k ≤ 7 and n − k ≥ 1. Finally, we make a conjecture for the edge irregularity strength of CSk,n−k for k ≥ 8 and n − k ≥ 1.

Edge Irregularity Strength of Binomial Trees

For a simple graph G, a vertex labeling φ: V (G) → {1, 2, · · ·, k} is called k- labeling. The weight of an edge uv in G, denoted by wφ(uv), is the sum of the labels of end vertices u and v. A vertex k-labeling is defined to be an edge irregular k- labeling of the graph G if for every two different edges e and f, wφ(e) wφ(f). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we prove that the edge irregularity strength of corona product of a tree T with K1 is es(T K1) = 2es(T). Further, we prove that the edge irregularity strength of binomial trees Bk is 2k−1, for k ≥ 1.

Extensions of results on phylogeny graphs of degree bounded digraphs

An acyclic digraph in which every vertex has indegree at most i and outdegree at most j is called an (i,j) digraph for some positive integers i and j. The phylogeny graph of a digraph D has V(D) as the vertex set and an edge uv if and only if one of the following is true: (u,v)∈A(D); (v,u)∈A(D); (u,w)∈A(D) and (v,w)∈A(D) for some w∈V(D). A graph G is a phylogeny graph (resp. an (i,j) phylogeny graph) if there is an acyclic digraph D (resp. an (i,j) digraph D) such that the phylogeny graph of D is isomorphic to G. Lee et al. (2017) and Eoh and Kim (2021) studied the (2,2) phylogeny graphs, (1,j) phylogeny graphs, (i,1) phylogeny graphs, and (2,j) phylogeny graphs. Their work was motivated by problems related to evidence propagation in a Bayesian network for which it is useful to know which acyclic digraphs have chordal moral graphs (phylogeny graphs are called moral graphs in Bayesian network theory). In this paper, we extend their work by giving necessary conditions of chordal (i,2) phylogeny graphs. We go further to give necessary conditions of (i,j) phylogeny graphs by listing forbidden induced subgraphs.

Pelabelan Super Graceful pada Graf B dan Graf serta Implementasinya pada Matlab

Suppose G is a graph with vertices p dan edges q. Super graceful labeling is a one-to-one function maping on f:V(G) È E(G)  {1,2, … , p + q} such that f(uv) = ½f(u) – f(v)½ differs for each edge uv Î E(G). A graph G is called super graceful if it can be labeled according to the definition of super graceful labeling. The graph B(m,n,k) is a graph consisteing of a path graph of length k conneting the star graph K(1,m) and K(1,n) at the pendant ends. Meanwhile, the graph Pn(1,2,…,n) is a path graph of length by n combining each vertex on the path graph with edges i to i members (1,2,…,n). In this journal, it will be shown that graph B(m,n,k) and graph Pn(1,2,…,n) are super graceful graphs.

On the total edge irregularity strength of certain classes of cycle related graphs

For a graph G=(V,E), an edge irregular total k-labeling is a labeling of the vertices and edges of G with labels from the set {1, 2, ..., k } such that any two different edges have distinct weights. The sum of the label of edge uv and the labels of vertices u and v determines the weight of the edge uv. The smallest possible k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G. We determine the exact value of the total edge irregularity strength for some cycle related graphs.

On r-dynamic Coloring of Ladder Graph and Tadpole Graph Using m-shadow Operation

An r-dynamic coloring is a proper k-coloring of a graph G = {V,E} such that the neighbors of every vertex v ∈ V(G) is colored using ς: V(G) → S(c) where S(c) is a set of colors. The coloring is made in such a way that it satisfies the conditions: (i) For any edge uv ∈ E(G), the color of u and color of v are distinct and (ii) The cardinality of coloring the neighbors of any vertex v should be greater than or equal to min{r, d(vG)}, where d(vG) is the degree of the vertex v. In this paper the lower bounds for the r-dynamic coloring of m-shadow graph of ladder graph Dm(Ln) and tadpole graph Dm(Ln,p) are attained. Using the lower bounds the exact solution of r-dynamic chromatic number of the ladder graph Ln and tadpole graph Ln,p by m-shadow operation are obtained.

Algorithmic Complexity and Bounds for Domination Subdivision Numbers of Graphs

Let G=V,E be a simple graph. A subset D⊆V is a dominating set if every vertex not in D is adjacent to a vertex in D. The domination number of G, denoted by γG, is the smallest cardinality of a dominating set of G. The domination subdivision number sdγG of G is the minimum number of edges that must be subdivided (each edge can be subdivided at most once) in order to increase the domination number. In 2000, Haynes et al. showed that sdγG≤dGv+dGv−1 for any edge uv∈EG with dGu≥2 and dGv≥2 where G is a connected graph with order no less than 3. In this paper, we improve the above bound to sdγG≤dGu+dGv−NGu∩NGv−1, and furthermore, we show the decision problem for determining whether sdγG=1 is NP-hard. Moreover, we show some bounds or exact values for domination subdivision numbers of some graphs.

Simultaneous Growth Strategy of High-Optical-Efficiency GaN NWs on a Wide Range of Substrates by Pulsed Laser Deposition.

Here, we explore a catalyst-free single-step growth strategy that results in high-quality self-assembled single-crystal vertical GaN nanowires (NWs) grown on a wide range of common and novel substrates (including GaN, Ga2O3, and monolayer two-dimensional (2D) transition-metal dichalcogenide (TMD)) within the same chamber and thus under identical conditions by pulsed laser deposition. High-resolution transmission electron microscopy and scanning transmission electron microscopy (HR-STEM) and grazing incidence X-ray diffraction measurements confirm the single-crystalline nature of the obtained NWs, whereas advanced optical and cathodoluminescence measurements provide evidence of their high optical quality. Further analyses reveal that the growth is initiated by an in situ polycrystalline layer formed between the NWs and substrates during growth, while as its thickness increases, the growth mode transforms into single-crystalline NW nucleation. HR-STEM and corresponding energy-dispersive X-ray compositional analyses indicate possible growth mechanisms. All samples exhibit strong band edge UV emission (with a negligible defect band) dominated by radiative recombination with a high optical efficiency (∼65%). As all NWs have similar structural and optical qualities irrespective of the substrate used, this strategy will open new horizons for developing III-nitride-based devices.

Hardness transitions and uniqueness of acyclic colouring

For k∈N, a k-acyclic colouring of a graph G is a function f:V(G)→{0,1,…,k−1} such that (i) f(u)≠f(v) for every edge uv of G, and (ii) there is no cycle in G bicoloured by f. For k∈N, the problem k-Acyclic Colourability takes a graph G as input and asks whether G admits a k-acyclic colouring. Ochem (EuroComb 2005) proved that 3-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree 4. Mondal et al. (2013) proved that 4-Acyclic Colourability is NP-complete for graphs of maximum degree five. We prove that for k≥3, k-Acyclic Colourability is NP-complete for bipartite graphs of maximum degree k+1, thereby generalising the NP-completeness result of Ochem, and adding bipartiteness to the NP-completeness result of Mondal et al.. In contrast, k-Acyclic Colourability is polynomial-time solvable for graphs of maximum degree at most 0.38k3/4. Hence, for k≥3, the least integer d such that k-Acyclic Colourability in graphs of maximum degree d is NP-complete, denoted by La(k), satisfies 0.38k3/4<La(k)≤k+1. We prove that for k≥4, k-Acyclic Colourability in d-regular graphs is NP-complete if and only if La(k)≤d≤2k−3. We also show that it is coNP-hard to check whether an input graph G admits a unique k-acyclic colouring up to colour swaps (resp. up to colour swaps and automorphisms).

Extremal results for [formula omitted]-free signed graphs

In this paper, we study a Turán-like problem in the context of signed graphs. Suppose that G˙ is a connected unbalanced signed graph of order n with e(G˙) edges and e−(G˙) negative edges, and let ρ(G˙) be the spectral radius of G˙. The signed graph G˙s,t (s+t=n−2) is obtained from an all-positive clique (Kn−2,+) with V(Kn−2)={u1,…,us,v1,…,vt} (s,t≥1) and two isolated vertices u and v by adding a negative edge uv and positive edges uu1,…,uus,vv1,…,vvt. Firstly, we prove that if G˙ is C3−-free, then e(G˙)≤n(n−1)2−(n−2), with equality holding if and only if G˙ is switching equivalent to G˙s,t. Secondly, we prove that if G˙ is C3−-free, then ρ(G˙)≤12(n2−8+n−4), with equality holding if and only if G˙ is switching equivalent to G˙1,n−3.

Bounding the Mostar index

Došlić et al. defined the Mostar index of a graph G as Mo(G)=∑uv∈E(G)|nG(u,v)−nG(v,u)|, where, for an edge uv of G, the term nG(u,v) denotes the number of vertices of G that have a smaller distance in G to u than to v. They conjectured that Mo(G)≤0.148‾n3 for every graph G of order n. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter Mo⋆(G)=∑uv∈E(G)(n−min⁡{dG(u),dG(v)}). For a graph G of order n, they show that Mo⋆(G)≤524(1+o(1))n3.We improve this bound to Mo⋆(G)≤(23−1)n3, which is best possible up to terms of lower order. Furthermore, we show that Mo⋆(G)≤(2(Δn)2+(Δn)−2(Δn)(Δn)2+(Δn))n3 provided that G has maximum degree Δ.

Analytic odd mean labeling of union and identification of some graphs

A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f∗ : E → Z such that for each edge uv with f(u) &lt; f(v), is injective. Clearly the values of f∗ are odd. We say that f is an analytic odd mean labeling of G. In this paper, we show that the union and identification of some graphs admit analytic odd mean labeling by using the operation of joining of two graphs by an edge.

Adjacent vertex distinguishing edge choosability of 1-planar graphs with maximum degree at least 23

For a simple graph G, an adjacent vertex distinguishing edge k-coloring of G is a mapping ϕ: E(G)→{1,2,…,k} such that ϕ(e1)≠ϕ(e2) for every adjacent edges e1 and e2 in E(G) and Cϕ(u)≠Cϕ(v) for each edge uv∈E(G), where Cϕ(v) (or Cϕ(u)) denotes the set of colors assigned to the edges incident with v (or u). For each edge e∈E(G), let L(e) be a list of possible colors that can be used on e. If, whenever we give a list assignment L={L(e)||L(e)|=k,e∈E(G)}, there exists an adjacent vertex distinguishing edge k-coloring ϕ such that ϕ(e)∈L(e) for each edge e∈E(G), then we say that ϕ is a list adjacent vertex distinguishing edge k-coloring. The smallest k for which such a coloring exists is called the adjacent vertex distinguishing edge choosability of G, denoted by cha′(G). A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. There is almost no result yet about cha′(G) if G is a 1-planar graph. We prove that cha′(G)≤Δ+3 for every 1-planar graph G with maximum degree Δ≥23.

On domination in signed graphs

Abstract In this article the concept of domination in signed graphs is examined from an alternate perspective and a new definition of the same is introduced. A vertex subset D of a signed graph S is a dominating set, if for each vertex v not in D there exists a vertex u ∈ D such that the sign of the edge uv is positive. The domination number γ (S) of S is the minimum cardinality among all the dominating sets of S. We obtain certain bounds of γ (S) and present a necessary and su cient condition for a dominating set to be a minimal dominating set. Further, we characterise the signed graphs having small and large values for domination number.