A modified anti-magic graph labeling for middle graph and total graph related to path

We investigate the generalized line graph, total graph and middle graph of the given graphs. First, replace a given graph G with a multigraph M having the same point set as G, then it constructs the total graph T(G) from G. Two suitable subgraphs Mx(G), Lx(G), of Tx(G) are considered as extensions of middle graph M(G) and line graph L(G) respectively.We also consider the connectivities of graphs which are obtained as a special case of Lx(G), Mx(G) and Tx(G). Then we get some extensions of established theorems.

An antimagic labeling of a graph with p vertices and q edges is a bijection from the set of edges to the set of integers {1, 2, . . . , q} such that all vertex weights are pairwise distinct, where a vertex weight is the sum of labels of all edges incident with the vertex. A graph is antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that that every connected graph, except K2, is antimagic. Recently, using completely separating systems, Phanalasy et al. showed that for each \({k\geq 2,\,q\geq\binom{k+1}{2}}\) with k|2q, there exists an antimagic k-regular graph with q edges and p = 2q/k vertices. In this paper we prove constructively that certain families of Cartesian products of regular graphs are antimagic.

Sparse graphs with vertex antimagic edge labelings

Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number chi _d(G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.

Antimagic graph labeling is the allotment of positive numbers to the edges of a graph such that the sum of graph edge labels incident to each node varies. The paper emphasizes on strong antimagic graph labeling for splitting graph and shadow graph of cycle. Modified theorems and their proofs on splitting and shadow graphs of cycle have been presented. The validity of the theorems has been illustrated through numerous examples.

The paramount is ensuring future-proof and robust cybersecurity in the era of quantum computing. According to graph theory, the mathematical concept of Edge Antimagic Labelling (EATL) is leveraged by introducing a theoretical approach for encryption. Based on sum own label, a uniquely labelled edge is each graph edge. This work presents a quantum-resistant and secure encryption framework to exploit the combinational complexity, uniqueness, and inherent non-linearity of EATL. Construct the Quantum safe graphs as a special class of graphs, and the cryptographic key is generated to use the antimagic labelling that derives the dynamic edge weights. Both quantum and classical threats counter to improve the structural masking and key randomness by proposing improved security of EATL. Based on label prediction and graph isomorphism, a numerically hard problem space is provided via graph graph-theoretic foundation. The post-quantum, secure, and lightweight cryptographic models are designed with the powerful tool of EATL, established using graph theory.

Irwing and Manlove instigate the knowledge of b -colorings and provided some inequalities for extreme bounds to explore the parameter on b -coloring. He denoted m-degree m(G) to determine the b -chromatic number. Here the parameter on b-coloring of Line graph of Tadpole Graph, Middle graph and Total graph of tadpole graph are being investigated. Also comparison of parameters on coloring and b-coloring has been done.

This paper presents algorithms , formulas and proofs for antimagic labelings for some graphs which are related in structure with helms. The labelings obtained are not only antimagic but also exhibit much regularity of patterns in the behaviour of the induced vertex labels.

In this paper, we studied the Incident Binary Equivalent Decimal Edge Graceful Labeling (IBEDE) [1] and Strong Incident Binary Equivalent Decimal Edge Graceful Labeling (SIBEDE) [1] for Wheel Graph W1,n and Middle Graph of Wheel Graph M(W1,n). Also we obtained the Wheel Graph W1,n and its Middle Graph M(W1,n) for n ≥ 3 are IBEDE Graceful Labeling . And the Wheel Graph W1,n for n > 3and its Middle Graph M(W1,n) for n ≥ 3 are SIBEDE Graceful Labeling. Here the Ramsey number of IBEDE and SIBEDE Graceful Labeling for W1,5 is 3 and the Ramsey number of IBEDE and SIBEDE Graceful Labeling for M(W1,3) is 18.

Dynamic coloring of a graph G is a proper coloring. The chromatic number of a graph G is the minimum k such that G has a dynamic coloring with k colors. In this paper we investigate the dynamic chromatic number for the Central graph, Middle graph, Total graph and Line graph of Web graph Wn denoted by C(Wn), M(Wn), T(Wn) and L(Wn) respectively.

For any non-trivial abelian group A under addition a graph $G$ is said to be $A$-textit{magic} if there exists a labeling $f:E(G) rightarrow A-{0}$ such that, the vertex labeling $f^+$ defined as $f^+(v) = sum f(uv)$ taken over all edges $uv$ incident at $v$ is a constant. An $A$-textit{magic} graph $G$ is said to be $Z_k$-magic graph if the group $A$ is $Z_k$ the group of integers modulo $k$. These $Z_k$-magic graphs are referred to as $k$-textit{magic} graphs. In this paper we prove that the total graph, flower graph, generalized prism graph, closed helm graph, lotus inside a circle graph, $Godotoverline{K_m}$, $m$-splitting graph of a path and $m$-shadow graph of a path are $Z_k$-magic graphs.

An antimagic labeling of a graph G is a bijection f:E(G)→{1,…,|E(G)|} such that the weights w(x)=∑y∼xf(y) distinguish all vertices. A well-known conjecture of Hartsfield and Ringel (1990) is that every connected graph other than K2 admits an antimagic labeling. For a set of distances D, a D-antimagic labeling of a graph G is a bijection f:V(G)→{1,…,|V(G)|} such that the weightω(x)=∑y∈ND(x)f(y) is distinct for each vertex x, where ND(x)={y∈V(G)|d(x,y)∈D} is the D-neigbourhood set of a vertex x. If ND(x)=r, for every vertex x in G, a graph G is said to be (D,r)-regular. In this paper, we conjecture that a graph admits a D-antimagic labeling if and only if it does not contain two vertices having the same D-neighborhood set. We also provide evidence that the conjecture is true. We present computational results that, for D={1}, all graphs of order up to 8 concur with the conjecture. We prove that the set of (D,r)-regular D-antimagic graphs is closed under union. We provide examples of disjoint union of symmetric (D,r)-regular that are D-antimagic and examples of disjoint union of non-symmetric non-(D,r)-regular graphs that are D-antimagic. Furthermore, lastly, we show that it is possible to obtain a D-antimagic graph from a previously known distance antimagic graph.

One important problem in graph theory is graph coloring or graph labeling.Labeling problem is a well-studied problem due to its wide applications, especially in frequency assignment in(mobile) communication systems, coding theory, x-ray crystallography, radar, circuit design, etc.For two non-negative integers, labeling of a graph is a function from the node set to the set of non-negative integers.For any two distinct vertices and of , a radio labeling is an injective function : 0 such that , | | 1 where is the diameter of .The radio number of , is the maximum value of taken overall radio labelings of .This paper determines the radio number of double triangular snake graphs.

We have formalized the definition of graph labeling in terms of combinatorial configurations. We have investigated the connection between edge and vertex (a, d)–distance antimagic labelings with such well-known configurations as separating systems and magic rectangle sets. We have obtained a solution to the problem of construction of indicated labelings for some types of graphs and certain values of a and d.

An one-one correspondence function λ from V(G) ∪ E(G) to the set {1, 2, …, |V(G) | + |E(G) |} is a total labeling of a finite undirected graph G without loops and multiple edges, where |V(G) |and |E(G) | are the cardinality of vertex and edge set of G respectively. A perfectly antimagic total labeling is a totally antimagic total labeling whose vertex and edge-weights that are also pairwise distint. Perfectly antimagic total (PAT) graph is a graph having such labeling. The topic of discovering perfectly antimagic total labeling of some families of graphs is discussed in this paper. We also came up with certain conclusions about dual of a perfectly antimagic total graphs. Finally, we provided that the necessary and sufficient condition for a dual of a regular and irregular PAT graph to be a PAT graph.