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# The Bandsize of Some Known Graphs

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1. INTRODUCTION Let G be a finite, simple graph of order n, n > 2. A vertex-numbering, f of G is a one-to-one function f: V(G) ( {1, 2, …, n }. The numbers f(v) – f(u) for all uv ( E(G) are called the edge- differences of the vertex-numbering f of G. The bandsize, bs(G) of G is the minimum number of distinct edge- differences over all vertex-numberings of G. In this research, we determine the exact bandsize of certain simple known graphs. By construction, we are able to provide an upper bound for the bandsize of a graph. Aside from that, we are able to build a relationship between the bandsize and the minimum degree of a graph. In ...

... Case 2: Consider the pair of edge-differences 1 and 2 in C5 and follow the same step as in case 1. Also, in any case, we cannot have the bs(P) = 3 but we can see that bs(P) > 3.

Case 3: Consider the pair of edge-differences 2 and 3 in C5 and follow the same step as in case 1. Also, in any case, we cannot have the bs(P) = 3 but we can see that bs(P) > 3.

For the possibility that we can have the bs(C5) = 2 if we will just choose 5 arbitrary numbers from the numbers 1, 2, …, 10, we will still have at most 3 pairings just like above if we choose 5 consecutive numbers. But we will still have the similar argument just like in the 3 cases.

Thus, the bandsize of the Petersen graph must 4.

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This paper is a theoretical discussion of the band size of some known graphs. It presents explanation and illustration of the band size of different known graphs.

language | english | |

wordcount | 2631 (cca 7.5 pages) | |

contextual quality | N/A | |

language level | N/A | |

price | free | |

sources | 2 |

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... Case 2: Consider the pair of edge-differences 1 and 2 in C5 and follow the same step as in case 1. Also, in any case, we cannot have the bs(P) = 3 but we can see that bs(P) > 3.

Case 3: Consider the pair of edge-differences 2 and 3 in C5 and follow the same step as in case 1. Also, in any case, we cannot have the bs(P) = 3 but we can see that bs(P) > 3.

For the possibility that we can have the bs(C5) = 2 if we will just choose 5 arbitrary numbers from the numbers 1, 2, …, 10, we will still have at most 3 pairings just like above if we choose 5 consecutive numbers. But we will still have the similar argument just like in the 3 cases.

Thus, the bandsize of the Petersen graph must 4.

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