Sparse graphs with vertex antimagic edge labelings

Graph labeling is the assignment of integers to edges or vertices or both depending on some factors which is consequential in the field of graph theory. In this paper, anti-magic graph labeling of two graphs, middle graph and total graph of path graph have been mainly focused. Modified proofs of theorems on anti-magic graph labeling have been introduced here. Both of the theorems are applicable for strong anti-magic graph. The correctness of the proofs has been demonstrated through various examples.

Consider a unicyclic graph G with edge set E(G). Let f be a real-valued symmetric function defined on the Cartesian square of the set of all distinct elements of G’s degree sequence. A graphical edge-weight-function index of G is defined as If(G)=∑xy∈E(G)f(dG(x),dG(y)), where dG(x) denotes the degree a vertex x in G. This paper determines optimal bounds for If(G) in terms of the order of G and a parameter z, where z is either the number of pendent vertices of G or the matching number of G. The paper also fully characterizes all unicyclic graphs that achieve these bounds. The function f must satisfy specific requirements, which are met by several popular indices, including the Sombor index (and its reduced version), arithmetic–geometric index, sigma index, and symmetric division degree index. Consequently, the general results obtained provide bounds for several well-known indices.

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.

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.

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.

Cryptography plays a crucial role in securing information and communications in the face of advancing technologies. Asymmetric encryption, also known as public-key cryptography, plays a crucial role in cryptography. Unlike symmetric encryption, which uses a single key for both encryption and decryption, asymmetric encryption involves a pair of keys, namely a public key and a private key. Asymmetric cryptography is closely associated with the secure management of keys, addresses, and transactions within the blockchain ecosystem, especially in cryptocurrency platform. In this study,we present a novel concept known as rainbow vertex antimagic coloring. This concept extends the idea of rainbow vertex coloring by incorporating antimagic labeling. Let f : E(G) → {1, 2, . . . , |E(G)|} be a function, where the weight of a vertex v ∈ V (G) with respect to f is defined as wf (v) = Σe∈E(v) f(e). Here, E(v) denotes the set of edges incident to v. Thefunction f is termed a vertex antimagic edge labeling if it assigns distinct weights to each vertex. A path is termed a rainbow path if, for any vertices u and v, all internal vertices on the u − v path have distinct weights. The rainbow vertexantimagic connection number of a graph G, denoted by rvac(G), is defined as the minimum number of colors requiredin any rainbow coloring derived from rainbow vertex antimagic labelings of G. In this paper, we will obtain some newlemmas or theorems concerning rvac(G), and we will implement the obtained lemmas or theorems of RVAC on asymmetric cryptography technique.

The local antimagic labeling on a graph G with |V| vertices and |E| edges is defined to be an assignment f : E --> {1, 2,..., |E|} so that the weights of any two adjacent vertices u and v are distinct, that is, w(u)̸ ̸= w(v) where w(u) = Σe∈<sub>E(u)</sub> f(e) and E(u) is the set of edges incident to u. Therefore, any local antimagic labeling induces a proper vertex coloring of G where the vertex u is assigned the color w(u). The local antimagic chromatic number, denoted by χla(G), is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we present the local antimagic chromatic number of unicyclic graphs that is the graphs containing exactly one cycle such as kite and cycle with two neighbour pendants.

Let be a graph of order n. Let be a bijection. For any vertex , the neighbor sum is called the weight of the vertex and is denoted by where N(v) is the open neighborhood of If for any two distinct vertices and then f is called a distance antimagic labelling. If the graph G admits such a labelling, then G is said to be a distance antimagic graph. This study gives a distance antimagic labelling for tensor product of two complete graph and sufficient condition so that the tensor product of a regular graph and complete graph is a distance antimagic graph.

Let G be a graph of order n. A bijection $$f:VG \longrightarrow \{1, 2, \ldots , n\}$$ is said to be distance antimagic if for every vertex v the vertex weight defined by $$w_fv =\sum _{x\in Nv}fx$$ is distinct. The graph which admits such a labeling is called a distance antimagic graph. For a positive integer k, define $$f_{k}:VG \longrightarrow \{1+k,2+k, \ldots , n+k\}$$ by $$f_{k}x=fx+k $$. If $$w_{f_{k}}u\ne w_{f_{k}}v$$ for every pair of vertices $$u,v \in V$$, for any $$k\ge 0$$ then f is said to be an arbitrarily distance antimagic labeling and the graph which admits such a labeling is said to be an arbitrarily distance antimagic graph. In this paper, we provide arbitrarily distance antimagic labelings for $$rP_n$$, generalised Petersen graph $$Pn,k, n \ge 5$$, Harary graph $$H_{4,n}$$ for $$n\ne 6$$ and also prove that join of these graphs is distance antimagic.

A handicap distance antimagic labeling of a graph G = ( V , E ) with n vertices is a bijection f : V → {1, 2, …, n } with the property that f ( x i ) = i , the weight w ( x i ) is the sum of labels of all neighbors of x i , and the sequence of the weights w ( x 1 ), w ( x 2 ), …, w ( x n ) forms an increasing arithmetic progression. A graph G is a handicap distance antimagic graph if it allows a handicap distance antimagic labeling. We construct r -regular handicap distance antimagic graphs of order $n \equiv 0 \pmod{8}$ for all feasible values of r .

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a Bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label f +(x) = Σf(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we first show that a d-leg spider graph has d + 1 ≤ χla ≤ + 2. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has χla = 4 if not all legs are of odd length. No 3-leg spider with all odd leg lengths and χla = 5 is found. This provides partial solutions to the characterization of k-pendant trees T with χla (T) = k + 1 or k + 2. We conjecture that almost all d-leg spiders of size q that satisfy d(d + 1) ≤ 2(2q - 1) with each leg length at least 2 has χla = d + 1.

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we characterize $s$-bridge graphs with local antimagic chromatic number 2.