| Summary: | 碩士 === 中原大學 === 應用數學研究所 === 94 === A graph G with q edges is said to be harmonious if there is a injection f from the vertex set of G into the group of integers modulo q such that when each edge xy is assigned as the label f(x)+f(y)(modq),the resulting edge labels are all distinct.
A graph nCm+K1 is a graph which is obtained by joining all nodes of nCm to a further node called the center.We say Cm+K1 to be a wheel.
For n>=2, we say nCm+K1 to be a multiple-wheel.Specially, we call 2Cm+K1 a double-wheel.
Graham and Slone[5] have proved that wheels are harmonious.Since Petrie and Smith[7] have proved that double-wheels have graceful labeling, we are interested in the topic of harmonious labelings of multiple-wheels.
A gear is a graph which is obtained from a wheel by adding a vertex between every pair of adjacent vertices of the outer cycle. Since Ma and Feng[8] have proved that gears have graceful labeling,we are also interested in the topic of harmonious labelings of gears.
In this thesis, we first show that for all integers n(mod4) is not equal 0 and m>=3, nCm+K1 have harmonious labelings,and then we prove that all gears have harmonious labelings.
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