DISTANCE MAGIC LABELING USING ALGEBRAIC APPROACH
Let G be a graph with order n and diameter d. Let D?{0,1,2,…,d} and N_D (v)={u?V(G) ?|d(u,v)?D}, where v?V(G). A bijection l?V(G)?{1,2,…,n} is called D-distance magic labeling of G if there is a nonnegative integer k such that ?_(u?N_D (v))??l(u)?=k for every v?V(G). If D={1}, D-distance magic labe...
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| Format: | Final Project |
| Language: | Indonesia |
| Online Access: | https://digilib.itb.ac.id/gdl/view/44309 |
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| Institution: | Institut Teknologi Bandung |
| Language: | Indonesia |
| Summary: | Let G be a graph with order n and diameter d. Let D?{0,1,2,…,d} and N_D (v)={u?V(G) ?|d(u,v)?D}, where v?V(G). A bijection l?V(G)?{1,2,…,n} is called D-distance magic labeling of G if there is a nonnegative integer k such that ?_(u?N_D (v))??l(u)?=k for every v?V(G). If D={1}, D-distance magic labeling is called distance magic labeling. An eigen value of a graph G is an eigen value of its adjacency matrix.
The main research in this final project is the relationship between eigen values of a graph and the existence of distance magic labeling on that graph. The graphs under consideration are regular graphs, distance-regular graphs, strongly regular graphs, line graphs of regular graphs, and graphs obtained from products of two regular graphs. Products of graphs considered are strong product, cartesian product, lexicographic product, and kronecker product.
By using eigenvalue of strongly regular graphs, necessary and sufficient conditions of strongly regular graphs having distance magic labeling are obtained. By using a matrix that has particular properties, distance magic labelings of some products of graph are obtained. By using previous results on hypercubes and by considering hypercubes as distance-regular graphs, the existence of D-distance magic labelings on hypercubes is obtained.
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