The Multiplicative Wiener Index, 𝜋(𝐺), is equal to the product of distance between all the pairs of vertices of G. In this paper, we investigate the Multiplicative Wiener Index of some standard graphs, which satisfies Harmonic Mean labeling.

Mean labeling is one of the best-known labeling methods for graphs. Despite the large number of papers published on the subject of graph labeling, there are some particular formulas to be used by researchers to mean-label graphs. In this paper, we introduced the concepts of Heinz Quarter Mean labeling graphs. Heinz Quarter Mean labeling for some graphs like Path, Cycle, Comb, Star graph and Complete Graph is proven as Heinz Quarter Mean Graphs.

Let G be a graph with p vertices and q edges and an injective function where each is a odd Fibonacci number and the induced edge labeling are defined by and all these edge labeling are distinct is called Odd Fibonacci Stolarsky-3 Mean Labeling. A graph which admits a Odd Fibonacci Stolarsky-3 Mean Labeling is called a Odd Fibonacci Stolarsky-3 mean graph.

A decomposition Gi of G is called a linear decomposition or Arithmetic decomposition if each Gi is connected and |E(Gi)| = a+(i − 1)d, for all i = 1,2,3, …, n and a, d ∊ Z. The Arithmetic decomposition with a = 2 and d = 2 is known as Even Decomposition (ED) As the number of edges of sub graph of G are even, we symbolize ED as (G2, G4, …,G2n). A decomposition (P2, P4, P6, …, P2n) of a graph G is an Even Path decomposition (EPD) if |E(P2i)| = 2i for all i = 1,2,3, …, n. Clearly q = n(n+1). This paper deals with Even Path Decomposition (EPD) of Root square mean graphs. Here we use graph labeling technique in Decomposition of Root square mean Graphs.

The Forgotten index of a graph G is defined as F(G) = over all edges of ,where , are the degrees of the vertices u and v in , respectively. In this paper, we introduced Forgotten index of some standard Stolarsky-3 Mean Graphs.

In this paper, We investigate the Laplacian matrix for some standard graphs such as Triangular Balloon graph, , ,

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph G = ( V , E ). In this paper we introduce Harary Index of Power 3 Tree Mean graphs.

A graph G = (V,E) with p vertices and q edges is said to be a Root Square Mean graph if it is possible to label the vertices x∈V with distinct labels f (x) from 1,2,……….. q+1 in such a way that when each edge e = uv is labeled with f (e=uv) = ⌈√((〖f(u)〗^2+〖f(v)〗^2)/2)⌉ or ⌊√((〖f(u)〗^2+〖f(v)〗^2)/2)⌋, then the resulting edge labels are distinct. In this case f is called a Root Square Mean labeling of G . In this paper we investigate the K- Root Square Mean labeling of some Graphs

The Wiener index W(G) of a connected graph G is the sum of distances of all pairs of vertices of G.

The Wiener index W(G) of a connected graph G is the sum of distances of all pairs of vertices of G. In this paper we investigate the Wiener index for Hurdle graph and H-Graph .

The Harmonic index \(H(G)\) of a graph \(G\) is defined as the sum of weights \(\frac{2}{d(u)+d(v)}\) of all edges \(u v\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). In this paper, we introduce harmonic index of some root square mean graphs.

Mean Labeling Graphs first introduced by Somasundram.S and Ponraj .R in the year2003.Several authors began to investigate various methods of Mean labelling after then.In thepicture filtering procedure, Power-3 mean labelling is very essential. The Power-3 Meanedges are generalized in this study so that the edges have separate labels.

Let f:V(G) → {1,2,….....q+1} be an injective function. For a vertex labeling f, the induced edge labeling f (e=uv) is defined by, f(e=uv)=[ f(u)2+f(v)22 ] or [ f(u)2+f(v)22 ], then the edge labels are distinct and are from {1,2,….. q}. Then f is called a Root Square Mean labeling of G. The concept of Root square mean labeling was introduced by S.S.Sandhya, S.Somasundaram and S.Anusa. In this paper we investigate some results on Root Square Mean labelingof graphs. Also find out Super root square mean labelingof some special graphs.

In this Paper we introduced a new concept Unlike degree Wiener index (UDWI) for some graph structures. We study the unlike degree Wiener Index for few graph structures such as Star graph, Path, Comb, Hurdle Graph.

A graph G =(V, E) with p vertices and q edges is said to he a mean graph if it is possible to label the vertices x e V with distinct elements f (x) from 0, 1, 2,..., q in such a way that when each edge e = uv is labeled with f(u) + f(v)/2 is even and f(u) + f(v)+1/2 if f(u) + f(v) is odd, then the resulting edge labels are distinct. In this case,/ is called a mean labeling of G. In this paper we prove that triangular Ladder TLu , TLn , circle dot K-1, circle dot K-1, D(T-n)circle dot K-1, Qn circle dot K-1, D(Q(n)) circle dot Ki are mean graphs.

A function f is called a SuperHeronian Mean Labeling of a graph G=(V,E) with p vertices and q edges if there is a injection f:V(G)→{1,2,3,...,p+q} such that the induced edge labeling f*(e=uv) is defined by, f*(e)=  or   .Any graph which satisfies the Super Heronian Mean Labeling is called the Super Heronian Mean graph . In this paper we present Flag graph, Dumbellgraph, Polygonalsnake, Balloon of the Triangular snake graphs are Super Heronian Mean graphs.

L et G =(V,E) be a graph with p vertices and q edges. Let f : V(G) ® {0,1,2,…, k+(q-1)} be an injective function such that the induced edge labeling f(e = uv) is defined by f(e) = or is a bijection from E to {k, k+1, k+2,…,k+(q-1)}.Then f has a (k,1)- contra harmonic mean labeling. Any graph which admits a (k,1)- contra harmonic mean labeling is called a (k,1)-contra harmonic mean graph. In this paper we investigate the (k,1)- Contra Harmonic mean labeling for some path related graphs.