Total absolute difference edge irregularity strength of some families of 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) ≠ 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 some families of graphs.Publisher's Versio

Total vertex irregularity strength of interval graphs

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers (usually positive integers). For a simple graph G = (V, E) with vertex set V and edge set E, a labeling φ : V ∪E → {1, 2, ..., k} is called total k-labeling. The associated vertex weight of a vertex x ∈ V (G) under a total k-labeling φ is defined as wt(x) = φ(x)+ P y∈N(x) φ(xy) where N(x) is the set of neighbors of the vertex x. A total k-labeling is defined to be a vertex irregular total labeling of a graph G, if wt(x) 6= wt(y) holds for every two different vertices x and y of G. The minimum k for which a graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G, tvs(G). In this paper, total vertex irregularity strength of interval graphs is studied. In particular, an efficient algorithm is designed to compute tvs of proper interval graphs and bounds of tvs are presented for interval graphs.Publisher's Versio

Minimum-Weight Edge Discriminator in Hypergraphs

In this paper we introduce the concept of minimum-weight edge-discriminators in hypergraphs, and study its various properties. For a hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, a function $\lambda: \mathcal V\rightarrow \mathbb Z^{+}\cup\{0\}$ is said to be an {\it edge-discriminator} on $\mathcal H$ if $\sum_{v\in E_i}{\lambda(v)}>0$, for all hyperedges $E_i\in \mathcal E$, and $\sum_{v\in E_i}{\lambda(v)}\ne \sum_{v\in E_j}{\lambda(v)}$, for every two distinct hyperedges $E_i, E_j \in \mathcal E$. An {\it optimal edge-discriminator} on $\mathcal H$, to be denoted by $\lambda_\mathcal H$, is an edge-discriminator on $\mathcal H$ satisfying $\sum_{v\in \mathcal V}\lambda_\mathcal H (v)=\min_\lambda\sum_{v\in \mathcal V}{\lambda(v)}$, where the minimum is taken over all edge-discriminators on $\mathcal H$. We prove that any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H(v)\leq n(n+1)/2$, and equality holds if and only if the elements of $\mathcal E$ are mutually disjoint. For $r$-uniform hypergraphs $\mathcal H=(\mathcal V, \mathcal E)$, it follows from results on Sidon sequences that $\sum_{v\in \mathcal V}\lambda_{\mathcal H}(v)\leq |\mathcal V|^{r+1}+o(|\mathcal V|^{r+1})$, and the bound is attained up to a constant factor by the complete $r$-uniform hypergraph. Next, we construct optimal edge-discriminators for some special hypergraphs, which include paths, cycles, and complete $r$-partite hypergraphs. Finally, we show that no optimal edge-discriminator on any hypergraph $\mathcal H=(\mathcal V, \mathcal E)$, with $|\mathcal E|=n (\geq 3)$, satisfies $\sum_{v\in \mathcal V} \lambda_\mathcal H (v)=n(n+1)/2-1$, which, in turn, raises many other interesting combinatorial questions.Comment: 22 pages, 5 figure

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

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

SOME CARTESIAN PRODUCTS OF A PATH AND PRISM RELATED GRAPHS THAT ARE EDGE ODD GRACEFUL

Let $G$ be a connected undirected simple graph of size $q$ and let $k$ be the maximum number of its order and its size. Let $f$ be a bijective edge labeling which codomain is the set of odd integers from 1 up to $2q-1$. Then $f$ is called an edge odd graceful on $G$ if the weights of all vertices are distinct, where the weight of a vertex $v$ is defined as the sum $mod(2k)$ of all labels of edges incident to $v$. Any graph that admits an edge odd graceful labeling is called an edge odd graceful graph. In this paper, some new graph classes that are edge odd graceful are presented, namely some cartesian products of path of length two and some circular related graphs

TOTAL EDGE AND VERTEX IRREGULAR STRENGTH OF TWITTER NETWORK

Twitter data can be converted into a graph where users can represent the vertices. Then the edges can be represented as relationships between users. This research focused on determining the total edge irregularity strength (tes) and the total vertices irregularity strength (tvs) of the Twitter network. The value could be determined by finding the greatest lower bound and the smallest upper bound. The lower bound was determined by using the properties, characteristics of the Twitter network graph along with the supporting theorems from previous studies, while the upper bound is determined through the construction of the total irregular labeling function on the Twitter network. The results in this study are the tes(TW)=18 and tvs(TW)=16

NILAI TOTAL TAK TERATUR TITIK PADA GRAF HASIL KALI COMB Pm DAN C4 DENGAN m BILANGAN GENAP

Abstrak Tidak Tersedi

Advances in Discrete Applied Mathematics and Graph Theory

The present reprint contains twelve papers published in the Special Issue “Advances in Discrete Applied Mathematics and Graph Theory, 2021” of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs

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

ليكن G  رسما بيانيا  برؤوس p وحواف q و  دالة متباينة وشاملة , حيث k عدد صحيح موجب. إذا كانت تسمية الحافة المستحثة  معرفة ب   لكل  المتباينة, فان علامة التبويب  تدعى وضع علامات غير منتظمة على حافة فيبوناتشي الفردية ل G. الرسم البياني الذي يعترف بوضع علامات غير منتظمة لحافة فيبوناتشي الفردية يسمى الرسم البياني غير المنتظم لحافة فيبوناتشي الفردية. قوة عدم انتظام حافة فيبوناتشي الفردية هي الحد الأدنى k الذي يعترف G بوضع علامات غير منتظمة لحافة فيبوناتشي الفردية. في هذا البحث ، تم تحديد قوة عدم انتظام حافة فيبوناتشي الفردية لبعض الرسوم البيانية للتقسيمات الفرعية والرسوم البيانية التي تم الحصول عليها من تحديد الرأس.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