Irregular Labeling Research Articles

Face irregular evaluations of family of grids

Let G = ( V , E , F ) be a connected plane graph. Let ρ : α ( V ) ∪ β ( E ) ∪ γ ( F ) → { 1 , 2 , … , k } be a k-labeling where α , β , γ ∈ { 0 , 1 } and ( α , β , γ ) ≠ ( 0 , 0 , 0 ) . The weight of a face f under ρ is defined as Wt ( f ) = α ∑ v ∼ f ρ ( v ) + β ∑ e ∼ f ρ ( e ) + γ ρ ( f ) . Then ρ is called a face irregular labeling of type ( α , β , γ ) if Wt ( f ) ≠ Wt ( g ) for any two faces f ≠ g . The face irregular strength of G is the smallest integer k such that G admits a face irreg ular k-labeling. In this paper we determine the face irregular strength of the grid graph P n + 1 □ P m + 1 under a labeling of type ( α , β , γ ) .

Computing Edge Irregularity Strength of Star and Banana Trees Using Algorithmic Approach

After the Chartrand definition of graph labeling, since 1988 many graph families have been labeled through mathematical techniques. A basic approach in those labelings was to find a pattern among the labels and then prove them using sequences and series formulae. In 2018, Asim applied computer-based algorithms to overcome this limitation and label such families where mathematical solutions were either not available or the solution was not optimum. Asim et al. in 2018 introduced the algorithmic solution in the area of edge irregular labeling for computing a better upper-bound of the complete graph \(es(K_n)\) and a tight upper-bound for the complete \(m\)-ary tree \({es(T}_{m,h})\) using computer-based experiments. Later on, more problems like complete bipartite and circulant graphs were solved using the same technique. Algorithmic solutions opened a new horizon for researchers to customize these algorithms for other types of labeling and for more complex graphs. In this article, to compute edge irregular \(k\)-labeling of star \(S_{m,n}\) and banana tree \({BT}_{m,n}\), new algorithms are designed, and results are obtained by executing them on computers. To validate the results of computer-based experiments with mathematical theorems, inductive reasoning is adopted. Tabulated results are analyzed using the law of double inequality and it is concluded that both families of trees observe the property of edge irregularity strength and are tight for \(\left\lceil \frac{|V|}{2} \right\rceil\)-labeling.

Total Face Irregularity Strength of Certain Graphs

The edge k-labeling ψ of G is defined by a mapping from EG to a set of integers 1,2,…,k, where the integer weight assigned to the vertex x∈VG is given as wψx=∑ψxy, such that the sum is taken over every vertex of y∈VG that is adjacent to x and the integer weights of adjacent vertices must be distinct for all vertices with x≠y. An irregular assignment of G using atmost k labels which is considered to be a minimum k is defined as irregularity strength of a graph G and can be denoted as sG. There are also further works on familiar irregular assignments, such as edge irregular labelings, vertex irregular total labelings, edge irregular total labelings, and face irregular entire k-labelings of plane graphs. A plane graph can be defined as a graph that is embedded in the plane in which no two lines will be intersected. In a plane graph the number of regions present are called faces and we denote it as F. The concept of total face irregularity strength is defined by the motivation of irregular networks and entire irregular face k-labeling. In our paper, we have obtained a minimum bound for the total face irregularity strength of two-connected plane graphs like cycle-of-ladder, C-necklace graph, P-necklace graph, sibling tree, and triangular graph.

Modular irregularity strength of disjoint union of cycle-related graph

Let G = (V,E) be a graph with a vertex set V and an edge set E of G, with order n. Modular irregular labeling of a graph G is an edge k-labeling φ:E → {1, 2,…,k} such that the modular weight of all vertices is all different. The modular weight is defined by wtφ(u) = Σv∈N(u) φ(uv) (mod n). The minimum number k such that a graph G has modular irregular labeling with the largest label k is called modular irregularity strength of G. In this research, we determine the modular irregularity strength for a disjoint union of cycle graph, $ (mC_{n})=\frac{mn}{2}+1 $ for n ≡ 0 (mod 4), a disjoint union of sun graph, ms(m(Cn ⊙ K1))2 = ∞ for n and m even and ms(m(Cn ⊙ K1)) = mn otherwise, and a disjoint union of middle graph of cycle graph, ms(mM(Cn)) = ∞ for n and m both odd numbers and $(mM({C_n})) = {{mn} \over 2} + 1$ otherwise.

On edge irregularity strength of different families of graphs

Edge irregular mapping or vertex mapping h : V (G) −→ {1, 2, 3, ..., s} is a mapping of vertices in such a way that all edges have distinct weights. We evaluate weight of any edge by using equation wth(cd) =h(c)+h(d), ∀c, d ∈ V (G) and ∀cd ∈ E(G). Edge irregularity strength denoted by es(G) is a minimum positive integer used to label vertices to form edge irregular labeling. In this paper, we find exact value of edge irregularity strength of linear phenylene graph P Hn, Bn graph and different families of snake graph.

LOCAL IRREGULARITY POINT COLORING ON THE RESULT OF SUBDIVISION OPERATION OF HELM GRAPHS

One of the sub-chapters studied in graphs is local irregularity vertex coloring of graph. The based on definition of local irregularity vertex coloring of graph, as follow : (i)l : V (G) →{1, 2, 3, . . . , k} as a vertex irregular labeling and w : V (G) → N, for every uv ∈ E(G), w(u) ̸=w(v) with w(u) = Pv∈N(u)l(v) and (i) Opt(l) = min{max(li); li is a vertex irregular labeling}. The chromatic number of the local irregularity vertex coloring of G denoted by χlis(G), is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this article, discuss about local irregularity vertex coloring of subdivision by helm graph (Sg(Hn)).

A Note on Distance Irregular Labeling of Graphs

Let us consider a~simple connected undirected graph G = ( V , E ) . For a~graph G we define a~ k -labeling ϕ : V ( G ) → { 1 , 2 , … , k } to be a~distance irregular vertex k -labeling of the graph G if for every two different vertices u and v of G , one has w t ( u ) ≠ w t ( v ) , where the weight of a~vertex u in the labeling ϕ is w t ( u ) = ∑ v ∈ N ( u ) ϕ ( v ) , where N ( u ) is the set of neighbors of u . The minimum k for which the graph G has a~distance irregular vertex k -labeling is known as distance irregularity strength of G , it is denoted as d i s ( G ) . In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order 1 , friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

A Note on Distance Irregular Labeling of Graphs

Let us consider a~simple connected undirected graph \(G=(V,E)\). For a~graph \(G\) we define a~\(k\)-labeling \(\phi: V(G)\to \{1,2, \dots, k\}\) to be a~distance irregular vertex \(k\)-labeling of the graph \(G\) if for every two different vertices \(u\) and \(v\) of \(G\), one has \(wt(u) \ne wt(v),\) where the weight of a~vertex \(u\) in the labeling \(\phi\) is \(wt(u)=\sum\limits_{v\in N(u)}\phi(v),\) where \(N(u)\) is the set of neighbors of \(u\). The minimum \(k\) for which the graph \(G\) has a~distance irregular vertex \(k\)-labeling is known as distance irregularity strength of \(G,\) it is denoted as \(dis(G)\). In this paper, we determine the exact value of the distance irregularity strength of corona product of cycle and path with complete graph of order \(1,\) friendship graph, Jahangir graph and helm graph. For future research, we suggest some open problems for researchers of the same domain of study.

TOTAL EDGE IRREGULAR LABELING FOR TRIANGULAR GRID GRAPHS AND RELATED GRAPHS

Let be a graph with and are the set of its vertices and edges, respectively. Total edge irregular -labeling on is a map from to satisfies for any two distinct edges have distinct weights. The minimum for which the satisfies the labeling is spoken as its strength of total edge irregular labeling, represented by . In this paper, we discuss the tes of triangular grid graphs, its spanning subgraphs, and Sierpiński gasket graphs.

Further results on local inclusive distance vertex irregularity strength of graphs

Let G = ( V , E ) be a simple undirected graph. A labeling f : V ( G )→{1, …, k } is a local inclusive d -distance vertex irregular labeling of G if every adjacent vertices x , y ∈ V ( G ) have distinct weights, with the weight w ( x ), x ∈ V ( G ) is the sum of every labels of vertices whose distance from x is at most d . The local inclusive d -distance vertex irregularity strength of G , lidis ( G ) , is the least number k for which there exists a local inclusive d -distance vertex irregular labeling of G . In this paper, we prove a conjecture on the local inclusive d -distance vertex irregularity strength for d = 1 for tree and we generalize the result for block graph using the clique number. Furthermore, we present several results for multipartite graphs and we also observe the relationship with chromatic number.

Modular irregularity strength on some flower graphs

Let G = ( V ( G ), E ( G )) be a graph with the nonempty vertex set V ( G ) and the edge set E ( G ) . Let Z n be the group of integers modulo n and let k be a positive integer. A modular irregular labeling of a graph G of order n is an edge k -labeling φ : E ( G )→{1, 2, …, k } , such that the induced weight function σ : V ( G )→ Z n defined by σ(v) = Σ ( u∈N(v)) φ(uv) (mod n) for every vertex v ∈ V ( G ) is bijective. The minimum number k such that a graph G has a modular irregular k -labeling is called the modular irregularity strength of a graph G , denoted by m s ( G ) . In this paper, we determine the exact values of the modular irregularity strength of some families of flower graphs, namely rose graphs, daisy graphs and sunflower graphs.

Pelabelan Jarak Tak Teratur Titik Pada Graf Persahabatan Lengkap Diperumum

Graph labeling is the labeling of graph elements such as vertex, edge and both. distance vertex irregular labeling is a type of labeling resulting from the development of distance magic labeling and (a, b)-distance anti-magic labeling. Let , be a simple graph. The distance vertex irregular labeling of is a vertex labeling so that the weight of each vertex is different. The weight of is calculated based on the sum of vertices label in the set of neighboring vertex , namely Distance vertex irregularity strength of , denoted as d , is the smallest value of the largest label so that has a distance vertex irregular labeling. This study aims to construct a generalized complete friendship graph , determine the labeling function, determine the distance vertex irregularity strength then formulate and prove the theorem resulting from the labeling. The object of this research is to label each vertex on a generalized complete friendship graph. This research method is a literature study obtained through various sources. Based on the research results, it is known that the graph has distance vertex irregular labeling. For an integer m and n, , the labeling function of is and . Distance vertex irregularity strength of generalized complete friendship graph is .

Odd Fibonacci edge irregular labeling for some trees obtained from subdivision and vertex identification operations

Let G be a graph with p vertices and q edges and be an injective function, where k is a positive integer. If the induced edge labeling defined by for each is a bijection, then the labeling f is called an odd Fibonacci edge irregular labeling of G. A graph which admits an odd Fibonacci edge irregular labeling is called an odd Fibonacci edge irregular graph. The odd Fibonacci edge irregularity strength ofes(G) is the minimum k for which G admits an odd Fibonacci edge irregular labeling. In this paper, the odd Fibonacci edge irregularity strength for some subdivision graphs and graphs obtained from vertex identification is determined.

Total vertex product irregularity strength of graphs

Consider a simple graph $G$. We call a labeling $w:E(G)\cup V(G)\rightarrow \{1, 2, \dots, s\}$ (\textit{total vertex}) \textit{product-irregular}, if all product degrees $pd_G(v)$ induced by this labeling are distinct, where $pd_G(v)=w(v)\times\prod_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum number used to label the members of $E(G)\cup V(G)$. The minimum value of $s$ that allows some irregular labeling is called \textit{the total vertex product irregularity strength} and denoted $tvps(G)$. We provide some general bounds, as well as exact values for chosen families of graphs. Keywords: product-irregular labeling, total vertex product irregularity strength, vertex-distinguishing labeling.

Modular Irregular Labeling on Firecrackers Graphs

Let G= (V, E) be a graph order n and an edge labeling ψ: E→{1,2,…,k}. Edge k labeling ψ is to be modular irregular -k labeling if exist a bijective map σ: V→Zn with σ(x)= ∑yϵv ψ(xy)(mod n). The modular irregularity strength of G (ms(G))is a minimum positive integer k such that G have a modular irregular labeling. If the modular irregularity strength is none, then it is defined ms(G) = ∞. Investigating the firecrackers graph (Fn,2), we find irregularity strength of firecrackers graph s(Fn,2), which is also the lower bound for modular irregularity strength, and then we construct a modular irregular labeling and find modular irregularity strength of firecrackers graph ms(Fn,2). The result shows its irregularity strength and modular irregularity strength are equal.

Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs

A k-labeling from the vertex set of a simple graph G=(V,E) to a set of integers {1,2,…,k} is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo |E(G)|. The modular edge irregularity strength of a graph is known as the maximal vertex label k, minimized over all modular edge irregular k-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs Pn+Km¯ and Cn+Km¯ and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds.

Modular Irregular Labeling On Complete Graphs

Let G be a simple graph of n order. An edge labeling such that the weights of all vertex are different and elements of the set modulo n, are called a modular irregular labeling. The modular irregularity strength of G is a minimum positive integer k such that G have a modular irregular labeling. If the modular irregularity strength is none, then we called the modular irregularity strength of G is infinity. In this article, we determine the modular irregularity strength of complete graphs.

On The Edge Irregularity Strength of Firecracker Graphs F2,m

Let be a graph and k be a positive integer. A vertex k-labeling is called an edge irregular labeling if there are no two edges with the same weight, where the weight of an edge uv is . The edge irregularity strength of G, denoted by es(G), is the minimum k such that has an edge irregular k-labeling. This labeling was introduced by Ahmad, Al-Mushayt, and Bacˇa in 2014. An (n,k)-firecracker is a graph obtained by the concatenation of n k-stars by linking one leaf from each. In this paper, we determine the edge irregularity strength of fireworks graphs F2,m.

Local irregular vertex coloring of comb product by path graph and star graph

Let [Formula: see text] be a simple graph and connected. A function [Formula: see text] is called vertex irregular [Formula: see text]-labeling and [Formula: see text] where [Formula: see text] The function of [Formula: see text] is called local irregular vertex coloring if every [Formula: see text] and [Formula: see text] vertex irregular labeling}. The local irregular chromatic number is denoted by [Formula: see text] In this paper, we study local irregular vertex coloring of [Formula: see text], and [Formula: see text]

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.