On Completely k-Magic Regular Graphs
Let k be a positive integer. A graph G = (V (G), E(G)) is said to be k-magic if there is a function (or edge labeling) ` : E(G) → Zk \ {0}, where Z1 = Z, such that the induced function (or vertex labeling) ` + : V (G) → Zk, defined by ` +(v) = P uv∈E(G) `(uv), is a constant map, where the sum is tak...
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Archīum Ateneo
2015
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| Online Access: | https://archium.ateneo.edu/mathematics-faculty-pubs/51 https://archium.ateneo.edu/cgi/viewcontent.cgi?article=1050&context=mathematics-faculty-pubs |
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| Institution: | Ateneo De Manila University |
| Summary: | Let k be a positive integer. A graph G = (V (G), E(G)) is said to be k-magic if there is a function (or edge labeling) ` : E(G) → Zk \ {0}, where Z1 = Z, such that the induced function (or vertex labeling) ` + : V (G) → Zk, defined by ` +(v) = P uv∈E(G) `(uv), is a constant map, where the sum is taken in Zk. We say that G is c-sum k-magic if ` +(v) = c for all v ∈ V (G). The set of all c ∈ Zk such that G is c-sum k-magic is called the sum spectrum of G with respect to k. In the case when the sum spectrum of G is Zk, we say that G is completely k-magic. In this paper, we determine all completely 1-magic regular graphs. After observing that any 2-magic graph is not completely 2-magic, we show that some regular graphs are completely k-magic for k ≥ 3, and determine the sum spectra of some regular graphs that are not completely k-magic. |
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