Edge Irregular Total K-labeling Research Articles

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.

Total absolute difference edge irregularity strength of Tp-tree graphs

A total labeling ξ is defined to be an edge irregular total absolute difference k-labeling of the graph G if for every two different edges e and f of G there is wt(e) 6= wt(f) where weight of an edge e = xy is defined as wt(e) = |ξ(e) − ξ(x) − ξ(y)|. The minimum k for which the graph G has an edge irregular total absolute difference labeling is called the total absolute difference edge irregularity strength of the graph G, tades(G). In this paper, we determine the total absolute difference edge irregularity strength of the precise values for Tp-tree related graphs.

Total Edge Irregularity Strength of the Cartesian Product of Bipartite Graphs and Paths

For a simple graph G = (V (G), E(G)), a total labeling ∂ is called an edge irregular total k-labeling of G if ∂ : V (G) ∪ E(G) → {1, 2, . . . , k} such that for any two different edges uv and u'v' in E(G), we have wt∂(uv) not equal to wt∂(u'v') where wt∂(uv) = ∂(u) + ∂(v) + ∂(uv). The minimum k for which G has an edge irregulartotal k-labeling is called the total edge irregularity strength, denoted by tes(G). It is known that ceil((|E(G)|+2)/3) is a lower bound for the total edge irregularity strength of a graph G. In this paper we prove that if G is a bipartite graph for which this bound is tight then this is also true for Cartesian product of G with any path.

On the total irregularity strength of convex polytope graphs

A vertex (edge) irregular total k-labeling ? of a graph G is a labeling of the vertices and edges of G with labels from the set {1,2,...,k} in such a way that any two different vertices (edges) have distinct weights. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x, whereas the weight of an edge is the sum of label of the edge and the vertices incident to that edge. The minimum k for which the graph G has a vertex (edge) irregular total k-labeling is called the total vertex (edge) irregularity strength of G. In this paper, we are dealing with infinite classes of convex polytopes generated by prism graph and antiprism graph. We have determined the exact value of their total vertex irregularity strength and total edge irregularity strength.

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.

On the total edge irregularity strength of general uniform cactus chain graphs with pendant vertices

Let G(V, E) be a graph. Throughout this paper, we use the notions of edge irregular total k-labeling and total edge irregularity strength of G (tes (G)). We verify tes of general uniform cactus chain graphs having (n - 2)r pendant vertices and length r. The result obtained is as follows: for n ≥ 6.

On total edge and total vertex irregularity strength of pentagon cactus chain graph with pendant vertices

Let G(V, E) be a graph with a non-empty set of vertices V and a set of edges E. A total k-labeling f: V ∪ E → {1,2, …, k} is called an edge irregular total k-labeling if the weight of each edge is distinct, where the weight of an edge e = xy is wt(e) = f(xy) + f(x) + f(y). Whereas, f is a vertex irregular total k-labeling if the weight wt(x) ≠ wt(y) for two distinct vertices x, y of V(G) where the weight wt(x) = f(x) + ∑ υx∈E(G) f(υx).The minimum k for which G has an edge irregular total k-labeling is called the total edge irregularity strength (tes) of G. Further, the total vertex irregularity strength (tvs) of G is the minimum k for which G has a vertex irregular total k-labeling. In this paper, we find tes and tvs of heptagon cactus chain graph with pendants and get the results as follows: tes and tvs .

On the total edge irregularity strength of some copies of ladder graphs

Let G = (V(G), E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f: V(G) ∪ E(G) → {1,2, ⋯, k }. The edge weight uv under the labeling f is defined by wf(uv) = f(u) + f(uv) + f(v) and denoted by by wf(uv) and. A total k-labeling of G is called edge irregular if every two distinct edges have distinct weight. The total edge irregularity strength of G is denoted by tes(G) and defined by the minimum k such that G has an edge irregular total k-labeling. The labeling was introduced by Bača et al. in 2007. In this paper, we determine the total edge irregularity strength of some copies of ladder graphs.

On The Total Edge Irregularity Strength of Some Copies of Books Graphs

Let G = (V(G),E(G)) be a graph and k be a positive integer. A total k-labeling of G is a map f : V(G) ∪ E(G) → {1,2, …, k}. The edge weight uv under the labeling f is denoted by Wf (uv) and defined by wf (uv) = f(u) + f(uv) + f(v). A total k -labeling of G is called edge irregular if there are no two edges with the same weight. The total edge irregularity strength of G, denoted by tes (G), is the minimum k such that G has an edge irregular total k-labeling. The labeling was introduced by Bača, Jendroľ, Miller, and Ryan in 2007. In this paper, we determine the total edge irregularity strength of some copies of book graphs.

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.

Total Edge Irregularity Strength of Arithmetic Book Graphs

For any simple undirected graph G(V, E), a map f : V ⋃ E → {1, 2, …, k} such that for any different edges xy and x’y’ their weights are distinct is called an edge irregular total k-labeling. The weight of edge xy is defined as the sum of edge label of xy, vertex label of x and vertex label of y. The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G and is denoted by tes(G). In this paper, we determine the exact value of the total edge irregularity strength of odd arithmetic book graph Bn(C3, 5, 7, …2n+1) and even arithmetic book graph Bn(C4, 6, 8, …, 2n+2) of n sheets. We found that the tes of odd arithmetic book graph Bn(C3, 5, 7, …, 2n+1) of n sheets is equal to the ceiling function of and the tes of even arithmetic book graph Bn(C4, 6, 8, …, 2n+2) is equal to the ceiling function of .

On total edge irregularity strength of some cactus chain graphs with pendant vertices

Let G(V, E) be a graph. A function f from to the set {1, 2, …, k} is called an edge irregular total k-labeling of G if the weights of any two different edges ux and vy in E(G) satisfy where the weight wf (ux) is equal to the sum of label of x, label of u and label of edge ux. The total edge irregularity strength of the graph G, denoted by tes(G), is the minimum number k for which G has an edge irregular total k-labeling. In this paper, we investigate the exact value of tes of triangular cactus chain graph and para square cactus chain graph. We get the total edge irregularity strength of triangular cactus chain with length r and r + 1 pendant vertices as follows: . Further, the total edge irregularity strength of para square cactus chain graph with length r and r pendant vertices is .

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 .

On the Total Edge Irregularity Strength of Generalized Butterfly 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 λ: V(G) ∪ E(G) → {1, 2, …, k} 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. A generalized butterfly graph, BFn, obtained by inserting vertices to every wing with assumption that sum of inserting vertices to every wing are same then it has 2n + 1 vertices and 4n − 2 edges. In this paper, we investigate the total edge irregularity strength of generalized butterfly graph, BFn, for n > 2. The result is .

Total edge irregularity strength of (n,t)-kite graph

Let G(V, E) be a simple, connected, 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 labeling of vertices and edges of G in such a way that for any different edges e and f, weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The total edge irregularity strength of G, tes(G), is defined as the minimum k for which a graph G has an edge irregular total k-labeling. An (n, t)-kite graph consist of a cycle of length n with a t-edge path (the tail) attached to one vertex of a cycle. In this paper, we investigate the total edge irregularity strength of the (n, t)-kite graph, with n > 3 and t > 1. We obtain the total edge irregularity strength of the (n, t)-kite graph is tes((n, t)-kite) = .

On total edge irregularity strength of centralized uniform theta graphs

Let G=(V,E) be a simple connected and undirected graph. Let f:V∪E→{1,2,…,k} be a total labeling of G. The weight of an edge uv is defined by wf(uv)=f(u)+f(v)+f(uv). The labeling f is called an edge irregular total k-labeling if wf(uv)≠wf(u′v′) for any two distinct edges uv, u′v′. If G admits such a labeling, then the minimum k is called the total edge irregularity strength of G. In this paper we determine the total edge irregularity strength of centralized uniform theta graphs.

Total edge irregularity strength of accordion graphs

An edge irregular total k-labeling \(\varphi : V\cup E \rightarrow \{ 1,2, \dots , k \}\) of a graph \(G=(V,E)\) is a labeling of vertices and edges of G in such a way that for any different edges xy and \(x'y'\) their weights \(\varphi (x)+ \varphi (xy) + \varphi (y)\) and \(\varphi (x')+ \varphi (x'y') + \varphi (y')\) are distinct. The total edge irregularity strength, tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling. We have determined the exact value of the total edge irregularity strength of accordion graphs.

Total edge irregularity strength of disjoint union of double wheel graphs

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3,...,k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u'v' their weights f (u) + f (uv) + f (v) and f (u') + f (u'v') + f (v') are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of disjoint union of p isomorphic double wheel graphs and disjoint union of p consecutive non-isomorphic double wheel graphs.

Irregular labelings of helm and sun graphs

A vertex irregular total k-labeling of a (p,q)-graph G=(V,E) is a labeling ϕ:V∪E→{1,2,…,k} such that the weights of the vertices wt(v)=ϕ(v)+∑uv∈Eϕ(uv) are different for all vertices. The total vertex irregularity strength tvs(G) is the minimum k for which G has a vertex irregular total k-labeling. The labeling ϕ is an edge irregular total k-labeling if for any two distinct edges e1=u1v1 and e2=u2v2, one has wt(e1)≠wt(e2) where wt(e1)=ϕ(u1)+ϕ(v1)+ϕ(u1v1). The total edge irregularity strength tes(G) is the minimum k for which G has an edge irregular total k-labeling. In this paper we determine tes(G) where G is the generalized helm and tvs(G) where G is the generalized sun graph.

On Total Edge Irregularity Strength of Generalized Web Graphs and Related Graphs

Let G = (V, E) be a simple, connected and undirected graph with non empty vertex set V and edge set E. An edge irregular total k-labeling \({f: V(G)\cup E(G) \to \{ 1,2, \ldots, k \}}\) is a labeling of vertices and edges of G in such a way that for any different edges xy and x′y′ their weights f(x) + f(xy) + f(y) and f(x′) + f(x′y′) + f(y′) are distinct. A total edge irregularity strength of graph G, denoted by tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling. In this paper, we determine the exact value of the total edge irregularity strength of the generalized web graph W(n, m) and two families of related graphs.