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On square difference graphs. (English) Zbl 1272.05170

Summary: In graph theory number labeling problems play vital role. Let \(G\) = (\(V,E\)) be a (\(p, q\))-graph with vertex set \(V\) and edge set \(E\). Let \(f\) be a vertex valued bijective function from \(V (G)\to \{0, 1,\dots, p-1\}\). An edge valued function \(f^{\star}\) can be defined on \(G\) as a function of squares of vertex values. Graphs which satisfy the injectivity of this type of edge valued functions are called square graphs. Square graphs have two major divisions: they are square sum graphs and square difference graphs.
In this paper we concentrate on square difference or \(SD\) graphs. An edge labeling \(f^{\star}\) on \(E(G)\) can be defined as follows. \(f^{\star} (uv) = |(f(u))^{2}-(f(v))^{2}|\) for every \(uv\) in \(E(G)\). If \(f^{\star}\) is injective, then the labeling is said to be a \(SD\) labeling. A graph which satisfies \(SD\) labeling is known as a \(SD\) graph. We illuminate some of the results on number theory into the structure of \(SD\) graphs. Also,established some classes of \(SD\) graphs and established that every graph can be embedded into a \(SD\) graph.

MSC:

05C78 Graph labelling (graceful graphs, bandwidth, etc.)
05C20 Directed graphs (digraphs), tournaments
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