Edge Xy Research Articles

Computing the Edge Irregularity Strength of Some Classes of Grid Graphs

Let G be a simple graph. A function ϕ:V(G)→{1,2,…,k} a vertex k-labeling which assigns labels to the vertices of G. For any edge xy in G, we define the weight of this edge as wϕ(xy)=ϕ(x)+ϕ(y). If all the edge weights are distinct, then ϕ is termed as an edge irregular k -labeling of G. The smallest possible value of k for which the graph G possesses an edge irregular k-labeling is denoted as the edge irregularity strength of G and is represented as es(G). In this paper, we investigate the edge irregular k-labeling of some classes of grid graphs, namely rhombic graph Rmn, triangular graph Lmn and octagonal graph Omn. As by-product, we obtain their precise value of edge irregularity strength.

Some Bench Mark Results on Total Domination Subdivision Stable Graph

For a graph G, the total dominating set defined as a set of vertices in S such that all the vertices in V(G) has at least one neighbor in S, the least cardinality is noted as t(G). The total domination number of each and every graph while subdividing any edge xy of G is equal to the total domination number of G, which results in the total domination subdivision stable graph abbreviated as TDSS and the symbolic expression is Gtsd(xy). The research paper, we introduce TDSS and proposed conditions under which a graph is TDSS and not TDSS.

Strong Regular Domination in Litact Graphs

Strong regular domination in a litact graph is a novel domination parameter that has been introduced in this paper. A dominating set D⊆V(G) is known as Strong regular dominating set of G, if for each point x∈V(G)-D there is a vertex y∈D with an edge xy∈E(G) and deg⁡(x)≤deg⁡(y) and all vertices of 〈D〉 holds the equal degree. The lowest cardinality of such vertices of D is known as strong regular domination number of G which is represented by γ_str (G). The current study aims by taking strong regular domination on a litact graph m(G) denoted by γ_str [m(G)] and to obtain some bounds on γ_str [m(G)] in terms of various parameters of G such as vertices, edges, maximum degree, diameter and so on and also in terms of various domination parameters of G such as total domination of G, connected domination of G and so on. Furthermore, outcomes resembling those of Nordhaus-Gaddum were also obtained.

Globally Linked Pairs of Vertices in Generic Frameworks

A d-dimensional framework is a pair (G, p), where G=(V,E)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$G=(V,E)$$\\end{document} is a graph and p is a map from V to Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document}. The length of an edge xy∈E\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$xy\\in E$$\\end{document} in (G, p) is the distance between p(x) and p(y). A vertex pair {u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u,v\\}$$\\end{document} of G is said to be globally linked in (G, p) if the distance between p(u) and p(v) is equal to the distance between q(u) and q(v) for every d-dimensional framework (G, q) in which the corresponding edge lengths are the same as in (G, p). We call (G, p) globally rigid in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} when each vertex pair of G is globally linked in (G, p). A pair {u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u,v\\}$$\\end{document} of vertices of G is said to be weakly globally linked in G in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} if there exists a generic framework (G, p) in which {u,v}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\{u,v\\}$$\\end{document} is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a (d+1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(d+1)$$\\end{document}-connected graph G in Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^d$$\\end{document} and then show that for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document} it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^2$$\\end{document}, which gives rise to an algorithm for testing weak global linkedness in the plane in O(|V|2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(|V|^2)$$\\end{document} time. Our methods lead to a new short proof for the characterization of globally rigid graphs in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^2$$\\end{document}, and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.

Simulations of Cayley graphs of dihedral group

Let Γ be a finite group with identity element e and let S ⊆ Γ − {e} which is inverse-closed, i.e., S = S−1 := {s−1 : s ∈ S}. An undirected Cayley graph on a group Γ with connection set S, denoted by Cay(Γ, S), is a graph with vertex set Γ and edges xy for all pairs x,y ∈ Γ such that xy−1 ∈ S. The dihedral group of order 2n, denoted by D2n, is a group generated by two elements a and b subject to the three relations: an = e, b2 = e, and bab−1 = a−1. In this paper, we investigate the Cayley graphs of the dihedral group for some connection sets S satisfying |S| = 1, 2, 3 by doing simulations using Wolfram Mathematica.

ABSOLUTE MEAN GRACEFUL LABELING IN THE CONTEXT OF m-SPLITTING AND DEGREE SPLITTING GRAPHS

A graph G with q edges is said to be absolute mean graceful if there is a one-to-one function f from V (G) to the set {0,±1,±2,±3,...,±q} such that when each edge xy is assigned the label |f(x)−f(y)| 2 , then the resulting edge labels are distinct. In this paper, the absolute mean graceful labeling of m-splitting and degree splitting graphs of some graphs are investigated.

EDGE IRREGULAR REFLEXIVE LABELING ON MONGOLIAN TENT GRAPH (M_(m,3)) AND DOUBLE QUADRILATERAL SNAKE GRAPH

Let G be an undirected, connected, and simple graph with edges set E(G)and vertex set V(G). An edge irregular reflexive k-labeling f is one in which the label for each edge is an integer number {1,2,…, k_e} and the label for each vertex is an even integer number {0,2,4,…,2k_v}, k = max{ k_e,2k_v}. This type of labeling results in distinct weights for each edge. The weight of an edge xy in a graph G with labeling f, indicated by wt (xy), is the total of the labels on the vertex that are incident to the edge as well as the edge label. The minimum value k of the largest label in the graph G is referred to as res (G), which stands for the reflexive edge strength of the graph G. The topic of edge irregular reflexive k-labeling for mongolian tent graph (M_(m,n)) and double quadrilateral snake graph (D(Q_n )) will be discussed in this paper. The res (M_(m,n)),m≥2,n=3 has been obtained that is ⌈(5m-1)/3⌉ for 5m-1≢2,3 (mod 6) and ⌈(5m-1)/3⌉+1 for 5m-1≡2,3 (mod 6). Also the res (D(Q_n )),n≥2 has been obtained that is ⌈(7n-7)/3⌉ for 7n-7≢2,3 (mod 6) and ⌈(5m-1)/3⌉+1 for 7n-7≡2,3 (mod 6).

EDGE IRREGULAR REFLEXIVE LABELING ON ALTERNATE TRIANGULAR SNAKE AND DOUBLE ALTERNATE QUADRILATERAL SNAKE

Let G in this paper be a connected and simple graph with set V(G) which is called a vertex and E(G) which is called an edge. The edge irregular reflexive k-labeling f on G consist of integers {1,2,3,...,k_e} as edge labels and even integers {0,2,4,...,2k_v} as the label of vertices, k=max{k_e,2k_v}, all edge weights are different. The weight of an edge xy in G represented by wt(xy) is defined as wt(xy)= f (x)+ f (xy)+ f (y). The smallest k of graph G has an edge irregular reflexive k-labeling is called the reflexive edge strength, symbolized by res (G). In article, we discuss about edge irregular reflexive k-labeling of alternate triangular snake A(T_n ) and the double alternate quadrilateral snake DA(Q_n ). In this paper, the res of alternate triangular snake A(T_n ) , n≥3 has been obtained. That is ⌈(2n-1)/3⌉ for n even,2n-1≢2,3 (mod 6),⌈(2n-1)/3⌉+1 for n even,2n-1=2,3 (mod 6),⌈(2n-2)/3⌉ for n odd,2n-2≢2,3 (mod 6), and ⌈(2n-2)/3⌉+1 for n odd,2n-2=2,3 (mod 6). Then, the reflexive edge strength of double alternate quadrilateral snake DA (Q_n) ⌈ (4n-1 )/3⌉for n even, 4n - 1 ≠2,3 (mod 6), ⌈ (4n-1 )/3⌉+1 for n even, 4n - 1 = 2,3 (mod 6), ⌈ (4n-4)/3⌉ for n odd, 4n - 4 ≠2,3 (mod 6), and ⌈ (4n-4 )/3⌉+1 for n odd, 4n - 4 = 2,3 (mod 6).

List 4-colouring of planar graphs

This paper proves the following result: If G is a planar graph and L is a 4-list assignment of G such that |L(x)∩L(y)|≤2 for every edge xy, then G is L-colourable. This answers a question asked by Kratochvíl et al. (1998) [10].

Induced Turán problems and traces of hypergraphs

Let F be a graph. We say that a hypergraph H contains an induced BergeF if the vertices of F can be embedded to H (e.g., V(F)⊆V(H)) and there exists an injective mapping f from the edges of F to the hyperedges of H such that f(xy)∩V(F)={x,y} holds for each edge xy of F. In other words, H contains F as a trace.Let exr(n,BindF) denote the maximum number of edges in an r-uniform hypergraph with no induced Berge F. Let ex(n,Kr,F) denote the maximum number of Kr’s in an F-free graph on n vertices. We show that these two Turán type functions are strongly related.

On the study of Rainbow Antimagic Connection Number of Corona Product of Graphs

Given that a graph G = (V, E). By an edge-antimagic vertex labeling of graph, we mean assigning labels on each vertex under the label function f : V → {1, 2, . . . , |V (G)|} such that the associated weight of an edge uv ∈ E(G), namely w(xy) = f(x) + f(y), has distinct weight. A path P in the vertex-labeled graph G is said to be a rainbow path if for every two edges xy, x′y ′ ∈ E(P) satisfies w(xy) ̸= w(x ′y ′ ). The function f is called a rainbow antimagic labeling of G if for every two vertices x and y of G, there exists a rainbow x − y path. When we assign each edge xy with the color of the edge weight w(xy), thus we say the graph G admits a rainbow antimagic coloring. The rainbow antimagic connection number of G, denoted by rac(G), is the smallest number of colors induced from all edge weight of antimagic labeling. In this paper, we will study the rac(G) of the corona product of graphs. By the corona product of graphs G and H, denoted by G ⊙ H, we mean a graph obtained by taking a copy of graph G and n copies of graph H, namely H1, H2, ..., Hn, then connecting vertex vi from the copy of graph G to every vertex on graph Hi , i = 1, 2, 3, . . . , n. In this paper, we show the exact value of the rainbow antimagic connection number of Tn ⊙ Sm where Tn ∈ {Pn, Sn, Sn,p, Fn,3}.

On the Study of Rainbow Antimagic Connection Number of Comb Product of Friendship Graph and Tree

Given a graph G with vertex set V(G) and edge set E(G), for the bijective function f(V(G))→{1,2,⋯,|V(G)|}, the associated weight of an edge xy∈E(G) under f is w(xy)=f(x)+f(y). If all edges have pairwise distinct weights, the function f is called an edge-antimagic vertex labeling. A path P in the vertex-labeled graph G is said to be a rainbow x−y path if for every two edges xy,x′y′∈E(P) it satisfies w(xy)≠w(x′y′). The function f is called a rainbow antimagic labeling of G if there exists a rainbow x−y path for every two vertices x,y∈V(G). We say that graph G admits a rainbow antimagic coloring when we assign each edge xy with the color of the edge weight w(xy). The smallest number of colors induced from all edge weights of antimagic labeling is the rainbow antimagic connection number of G, denoted by rac(G). This paper is intended to investigate non-symmetrical phenomena in the comb product of graphs by considering antimagic labeling and optimizing rainbow connection, called rainbow antimagic coloring. In this paper, we show the exact value of the rainbow antimagic connection number of the comb product of graph Fn⊳Tm, where Fn is a friendship graph with order 2n+1 and Tm∈{Pm,Sm,Brm,p,Sm,m}, where Pm is the path graph of order m, Sm is the star graph of order m+1, Brm,p is the broom graph of order m+p and Sm,m is the double star graph of order 2m+2.

Lexicographic palindromic products

A graph G on n vertices is palindromic if there is a vertex-labeling bijection f : V(G) → {1, 2, …, n} with the property that for any edge vw ∈ E(G), there is an edge xy ∈ E(G) for which f(x) = n − f(v) + 1 and f(y) = n − f(w) + 1. We characterize the conditions under which a lexicographic product G ∘ H is palindromic, as follows: If G ∘ H has an even number of vertices, then it is palindromic if and only if at least one of G or H is palindromic. If G ∘ H has an odd number of vertices, then it is palindromic if and only if both G and H are palindromic.

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge set EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called a total edge irregular labeling on Γ if for any two different edges xy and x′y′ in EΓ the numbers α(x)+α(xy)+α(y) and α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that Γ can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph Γ. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings.

A REMARK ON THE EDGE IRREGULARITY STRENGTH OF CORONA PRODUCT OF TWO PATHS

With respect to a simple graph G, a vertex labeling ϕ: V(G) > {1,2,...,k) is known as k-labeling. The weight corresponding to an edge xy in G, expressed as wϕ (xy), represents the labels sum of end vertices x and y, given by wϕ (xy) = ϕ(x) + ϕ(y) A vertex k-labeling is expressed as an edge irregular k-labeling with respect to graph G provided that for every two distinct edges e and f, there exists wϕ(e) ≠ wϕ(f) Here, the minimum k where the graph G possesses an edge irregular k-labeling is known as the edge irregularity strength with respect to G, expressed as (G). Here, we examine the edge irregularity strength’s exact value of corona product with respect to two paths Pn and Pm , in which n ≥ 2 and m = 3, 4, 5.

Edge Irregular Reflexive Labeling for Some Classes of Plane Graphs

For a graph G, we define a total k-labeling ϕ as a combination of an edge labeling ϕe(x) → {1, 2, . . . , ke} and a vertex labeling ϕv(x) → {0, 2, . . . , 2kv}, such that ϕ(x) = ϕv(x) if x ∈ V (G) and ϕ(x) = ϕe(x) if x ∈ E(G), where k = max {ke, 2kv}. The total k-labeling ϕ is called an edge irregular reflexive k-labeling of G, if for every two edges xy, x0y0of G, one has wt(xy) 6=wt(x0y0), where wt(xy) = ϕv(x) + ϕe(xy) + ϕv(y). The smallest value of k for which such labeling exists is called a reflexive edge strength of G. In this paper, we study the edge irregular reflexive labeling on plane graphs and determine its reflexive edge strength.

Elegant labeling of sun graphs and helm graphs

Let G be a simple and finite graph having q edges. An elegant labeling f of G is an injective function from the set of vertices of G to the set {0,1,2,…,q} such that the induced edge labels, where each edge xy is assigned the label f*(xy) = f(x) + f(y) (mod(q + 1)), are distinct and non zero. If a graph can be labeled by an elegant labeling, then the graph is said to be elegant. In this paper we show that sun graph n-sun and helm graph Hn , where n is an odd integer at least 3, are elegant.

Q8 difference cordial labeling

Let Q8 be a quaternion group. Let G=(V,E) be a graph. Let f: V(G)→Q 8 . For each edge xy assign the label 0 when |o(f(x))−o(f(y))|=0 and 1 otherwise. The function f is called Q 8 cordial difference labeling of G if |v f (x)−v f (y)|≤1 and |e f (0)−e f (1)|≤1, where v f (x), v f (y) denote the total number of vertices labeled with x, y in Q 8 and e f (0), e f (1) denote the total number of edges labeled with 0,1 respectively. A graph G which admits a group Q 8 difference cordial labeling is called Q 8 difference cordial graph. In this paper, we prove the existence of this labeling to the graphs viz., path, ladder related graphs and snake related graphs. Keywords: group Q 8 cordial; cordial labeling; quaternion group labeling.

Total Edge Irregularity Strength of q Tuple Book Graphs

Let G(V, E) be a simple, undirected, and finite graph with a vertex set V and an edge set E. An edge irregular total k-labelling is a function f from the set V \cup E to the set of non-negative integer set (1, 2, ... , k) such that any two different edges in E have distinct weights. The weight of edge xy is defined as the sum of the label of vertex x, the label of vertex y and the label of edge xy. The minimum k for which the graph G can be labelled by an edge irregular total k-labelling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of triple book graphs, quadruplet book graphs and quintuplet book graphs. In this paper, we construct an edge irregular total of k-labelling and show the exact value of the total edge irregularity strength of q tuple book graphs.

Total edge irregularity strength of triple book graphs

Let G(V, E) be a finite, simple and undirected graph with a vertex set V and an edge set E. An edge irregular total k-labeling is a map f : V ⋃ E → {1, 2, …, k} such that for any two different edges xy and x′y′ in E, ω(xy) ≠ ω(x′y′) where ω(xy) = f(x) + f(y) + f(xy). The minimum k for which the graph G admits an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). We have constructed the formula of an edge irregular total k-labelling and determined the total edge irregularity strength of book graphs and double book graphs. In this paper, we construct an edge irregular total k-labeling that can be used for book graphs, double book graphs, and triple book graphs. We also show the exact value of the total edge irregularity strength of triple book graphs.