Abstract Graph Labeling is a significant area of graph theory that is used in a variety of applications like coding hypothesis, x-beam crystallography, radar, cosmology, circuit design, correspondence network tending to, and database administration. This study provides a general overview of graph naming in heterogeneous fields, however it primarily focuses on graph subdivision. The even vertex odd meanness of super subdivide of various graphs is discussed in this study. The graphs generated by super subdivided of path, cycle, comb, crown, and planar grid are even-odd mean graphs, according to our proof.

Abstract: Let G be a nontrivial connected graph. Then G is called a rainbow connected graph if there exists a coloring c : E(G) → {1, 2, ..., k}, k ∈ N, of the edges of G, such that there is a u − v rainbow path between every two vertices of G, where a path P in G is a rainbow path if no two edges of P are colored the same. The minimum k for which there exists such a k-edge coloring is the rainbow connection number rc(G) of G. If for every pair u, v of distinct vertices, G contains a rainbow u − v geodesic, then G is called strong rainbow connected. The minimum k for which G is strong rainbow-connected is called the strong rainbow connection number src(G) of G. The exact rc and src of the rotationally symmetric graphs are determined.

Abstract: For a graph G the duplication operation of a vertex v by a new edge e = uw results in a new graph G’ such that N (u) = {v, w} and N (w) = {v, u}. The duplication operation of an edge e = uv by a new vertex w results in a new graph G’’ such that N (w) = {u, v}. In this article we have discussed that the properties of a graph, with minimum degree 2 of any vertex, to be super vertex-antimagic total and to be super edge-antimagic total are invariant under the above duplication operations. Also, we have discussed on the existence of the so-called k super vertex-antimagic total vertex modifications and k super edge-antimagic total edge modifications for graphs.

Abstract A zero-sum flow is an assignment of nonzero integers to the edges such that the sum of the values of all edges incident with each vertex is zero, and we call it a zero-sum k-flow if the absolute values of edges are less than k. We recall the zero-sum flow number of G as the least integer k for which G admitting a zero sum k-flow. In this paper we gave complete zero-sum flow and zero sum numbers for Octagonal Grid and Generalized Prism.

The topological index of a molecular structure correlate certain physicochemical properties such as boiling point and stability of that molecular structure. Recently several topological indices have been introduced and studies have shown their association with certain physical and chemical properties of other molecules. Specifically, Zagreb, Atom-bound connectivity (ABC) and Sanskruti indices have been correlated with extent of branching, enthalpy and entropy of some organic molecules. In this study we have calculated Zagreb, Atom-bound connectivity (ABC) and Sanskruti indices (S) for Copper (I) oxide and Copper (II) oxide CuO. Additionally, from the experimentally available data, we calculated entropy and enthalpy per unit cell for both the oxides and compared with Atom-bound connectivity and Sanskruti indices. It appears that these indices have very poor relation with these physical properties in both oxides of copper and they may need some modifications to show considerable compliance as reported in other organic molecules."

Molecular descriptors are being widely used in QSAR/QSPR studies in chemistry and drug designing as well as modeling of compounds. Different topological descriptors have been formulated to investigate the physio chemical properties and chemical reactivity of compounds. In this article we gave exact relations for first and second Zagreb index, hyper Zagreb index, multiplicative Zagreb indices as well as first and second Zagreb polynomials for some benzenoid systems.