Coding and Descriptions for the Subset of Twin Star Mean Labeling

Graph labeling was first introduced by Rosa in 1966. Labeling of graphs is an assignment of nonnegative integers to vertices, edges or both according to some specified conditions. Mean labeling of graphs was introduced by Somasundaram and Ponraj in 2003. Subsequently, labelings of graphs were done with geometric mean, harmonic mean etc. In this paper we introduce two concepts called ’Logarithmic mean labeling’, ’Identric mean labeling’ and acquire their mean labelings for some standard graphs.

In graph theory, labeling of vertices and edges according to specific rules offers insight into the structural properties of graphs and their potential applications. This study focuses on a particular type of graph labeling known as Super Lehmer-3 Mean Labeling. A graph is said to admit such a labeling if there exists an injective vertex labeling such that the corresponding edge labels, derived from the Lehmer-3 mean of their incident vertex labels, are also distinct and satisfy certain arithmetic conditions. Graphs that admit such a labeling are termed Super Lehmer-3 Mean Graphs.In this work, we investigate the existence of super Lehmer-3 mean labeling in various families of theta-related graphs. Specifically, we analyze standard theta graphs and their structural variations obtained by subdivisions and attachments of pendant vertices. The results confirm that these modified forms of theta graphs also admit super Lehmer-3 mean labeling. Additionally, we extend our exploration to harmonic mean labeling in selected theta graphs, highlighting the conditions under which this alternative labeling scheme is valid.The findings presented in this paper contribute to the growing body of knowledge in graph labeling theory, offering new characterizations and constructions of labeled graph families. These results may have implications for theoretical studies and applications where such labeling structures are relevant.

The concept of one modulo three mean labeling graph is, if there is an injective function φ from the vertex set of G to the set {a/0 ≤ a ≤ 3q − 2 and either a ≡ 0(mod3) or a ≡ 1 (mod3)} where q is the number of edges of G and φ induces a bijection φ* from the edge set of G to {a/1≤ a ≤ 3q − 2, a ≡ 1 (mod3) } given by and the function φ is called one modulo three mean labeling of G. In this paper, we obtain the results of one modulo three mean labeling of some several graphs.

In this paper, we introduce a coding technique for converting a text message using sub super mean labeling on two and three star graphs to a picture coding message. The first part of this paper there are some definitions and discussions for the existence graph and in the second part there are some illustrations to explain how the text message is converted in to picture code, by using graph labeling, and it maintain secrecy, possess intricacy and provide a sense of sufficiency. Secrecy is the practice of hiding information from certain individuals or groups who do not have the need to know, perhaps while sharing it with other individuals that which is kept hidden is known as secret.

Let G be a (p,q)-graph and : V(G)→{k,k+1,k+2,k+3,…,p+q+k−1} be an injection. For each edge e=uv, let f*(e)=⌈ f(u)+f(v)2 ⌉. Then f is called a k-super mean labeling if f(V)∪{f*(e):e∈E(G)}={k,k+1,k+2,…,p+q+k−1}. A graph that admits a k-super mean labeling is called k-super mean graph. In this paper, we present k-super mean labeling of C2n(n≠2) and super mean labeling of Double cycle C(m,n), Dumb bell graph D(m,n) and Quadrilateral snake Qn.

Let G be a graph and f : V(G) → {1, 2, 3, …, p + q} be an injection. For each edge e = uv, the induced edge labeling f* is defined as follows: Then f is called super mean labeling if f(V(G)) ∪ {f*(e): e ∈ E(G)} = {1, 2, 3, …, p + q}. A graph that admits a super mean labeling is called super mean graph. In this paper we study the super meanness of subdivision graph of K 1,3, super meanness of Pτ,4, super meanness of some caterpillars and super meanness of some duplicate graphs.

Let G(V,E) be a graph with the vertex set V and the edge set E, respectively. By a graph G=(V,E) we mean a finite undirected graph with neither loops nor multiple edges. The number of vertices of G is called order of G and it is denoted by p. Let G be a (p,q) graph. A super mean graph on G is an injection f:V→{1,2,3…,p+q} such that, for each edge e=uv in E labeled by f⁎e=fu+f(v)/2, the set fV∪{f⁎e:e∈E} forms 1,2,3,…,p+q. A graph which admits super mean labeling is called super mean graph. The total graph T(G) of G is the graph with the vertex set V∪E and two vertices are adjacent whenever they are either adjacent or incident in G. We have showed that graphs T(Pn) and TCn are super mean, where Pn is a path on n vertices and Cn is a cycle on n vertices.

On Super mean graphs

Let G(V, E) be a graph with p vertices and q edges and f: V(G) → {1,2,3,…, p + q} be an injection. For each edge e = uv, let . Then f is called a super mean labeling of G if {f (V(G)) ∪ {f *(e): e ∊ E(G)} = {1,2,3,…, p + q}. A graph that admits a super mean labeling is called a super mean graph. In this paper we prove that , , , T @ Pn, T @ 2Pn, are super mean graphs if T is a Tp -tree.

An even vertex odd mean labeling of an undirected graph with q edges is a one to one function f from the set of vertices V(G) to the set 0, 2, 4, … , 2q such that the induced function f* from E(G) to the set 1, 3, 5, … , 2q – 1 defined by is a bijection. A graph that admits an even vertex odd mean labeling is called even vertex odd mean graph. In this paper, we prove that D 2 (Pn ), D 2 (Ln ), D 2 ([P 2n , Sm ]), S(Ln ), S(Pn ⊗ K 1 ), S(SLn ), S′(Pn ), S′(k 2,n ), S′(Pn ⊗ K 1 ), S′(Cn ) for n ≡ 0 (mod 4) and S′([P 2n , Sm ]) are even vertex odd mean graphs.

A graph G with |E(G)|=q, an injective function f : V (G) → {0, 2, 4, ..., 2q} is an even vertex odd mean labeling of G that induces the values {f(u)+f(v)}/{2} for the q pairs of adjacent vertices u,v are distinct. In this paper, we investigate an even vertex labeling for the calendula graphs. Moreover we introduce the definition of arbitrary calendula graph and prove that the arbitrary calendula graphs are also even vertex odd mean graphs.

A radio harmonic mean labeling of a connected graph G is a one to one map f from the vertex set V(G) to the set of natural numbers N such that for two distinct vertices u and v of G, d(u,v)+[ 2f(u)f(v)f(u)+f(v) ]≥1+diam(G). The radio harmonic mean number of f, rhmn(f) is the maximum number assigned to any vertex of G. The radio harmonic mean number of G, rhmn(G) is the minimum value of rhmn(f) taken over all radio harmonic mean labeling f of G. In this paper we have determined the radio harmonic mean number of some star related graphs.

A graph G = (V,E) with p vertices and q edges is said to be a Root Cube Even Mean Graph if it is possible to label the vertices x V with distinct elements f(x) from 1,2......,q+1 in such a way that when each edge e = uv is labeled with , then the resulting edge labels are distinct. Here f is called a Root Cube Even Mean Labeling of G. In this paper we prove that path, comb, Pn⊙K1,2 is a root cube even Mean Labeling graph.

We introduce a new type of labeling known as triangular mean labeling. Properties of this labeling are studied and triangular mean behaviors of some standard graphs are investigated.

Some results on super Heronian mean labeling of graphs