2. Resolvability of some families of Corona Product Graphs By Kareem Nawaz Supervised by Dr. Muhammad Imran

3. TABLE OF CONTENTS Introduction and Definitions Methodology Results and Discussions Conclusion References

4. What is Graph Definition − A graph denote as G=(V,E)G=(V,E)) consists of a non- empty set of vertices or nodes V and a set of edges E.

5. Graph In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

6. Resolvability (Graphs ) The ideas of resolvability and area in graphs were depicted autonomously by Harary and Melter [1], and Slater [2], to characterize a similar structure in a diagram. After these papers were distributed, a few creators created various hypothetical takes a shot at this theme. Slater portrayed the helpfulness of these thoughts into long range helps to route. Additionally, these ideas have a few applications in science for speaking to synthetic mixes [11,12] or to issues of example acknowledgment and picture preparing, some of which include the utilization of various leveled information structures.

7. Types of Graph

8. Linear Graph

9. Some types of Graph Simple Graph A simple graph is a graph which has no loop as well as no parallel edges. Weighted graph If a number (weight) is authorized to each edge. Such weights may signify, for example, costs, lengths or capacities, etc. depend on the problem. Null Graph A Graph having no edge is known as Null Graph

10. Complete Graph A simple graph in which an edge lies between each pair of vertices.

11. PRODUCT OF GRAPH

12. Cartesian product

14. LEXICOGRAPHIC PRODUCT GRAPH In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the Cartesian Product product V(G) × V(H); and any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent with x in G or u = x and v is adjacent with y in H.

15. LEXICOGRAPHIC PRODUCT GRAPH

16. Direct Product

18. Corona Product

19. See that G is associated if and just if G ⊙ H is associated. In addition, it is seen from the definition that this item is neither a commutative activity nor a cooperative. Figure 1.6 demonstrates a few instances of crown items and furthermore underscores the way that the crown item isn't commutative. The possibility of crown result of two charts was right off the bat proposed by Frucht and Harary. This item isn't commonplace and generally found. One of the reason ought to be the reality, that crown item is a straightforward activity on two charts and some scientific properties can be specifically results of its components. Inspite of this, it is fascinating to contemplate metric measurement related parameters in this item. Additionally, there are takes a shot at some topological files and the impartial chromatic number of corona product.

20. Results & Discussion

21. Theorem

22. Proof

23. Proof

24. Conclusion We have exhibited two calculations for illustration graphs. Both are quick and produce decent looking pictures. In the power coordinated technique each factor has a similar impact on the format, however it is as it were useble if the quantity of variables is little. By keeping edges short, it keeps away from edge intersections. For some factors, the edge intersections progress toward becoming unavoidable and it is useful to make a few edges in length and pack them. This methodology is available in the various leveled format of product graph, which additionally catches on the idea of the elements characterizing a chain of importance inside the item and permits an explorative examination of substantial graph products.

25. References 1. N. Alon, E. Lubetzky, The Shannon capacity of a graph and the independence numbers of its powers, Transactions on Information Theory 52 (5) (2006) 2172-2176. 2. M. Azari, A. Iranmanesh, Chemical graphs constructed from rooted product and their Zagreb indices, MATCH Communications in Mathematical and in Computer Chemistry 70 (3) (2013) 901-919. 3. R. F. Bailey, P. J. Cameron, Base size, metric dimension and other invariants of groups and graphs, Bulletin of the London Mathematical Society 43 (2) (2011)209-242. 4. R. Balakrishnan, P. Paulraja, Hamilton cycles in tensor product of graphs, Discrete Mathematics 186 (1-3) (1998) 1-13. 5. Z. Baranyai, G. R. Sz asz, Hamiltonian decomposition of lexicographic product, Journal of Combinatorial Theory, Series B 31 (3) (1981) 253-261.