Bounds on Sum Number in Graphs
نویسندگان
چکیده
A simple undirected graph G is called a sum graph if there is a labeling L of the vertices of G into distinct positive integers such that any two vertices u and v of G are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (H) of a graph H = (V; E) is the least integer r such that graph G consisting of H and r isolated vertices is a sum graph. It is clear that (H) jEj. In this paper, we discuss general upper and lower bounds on the sum number. In particular, we prove that the average of (H) over all graphs H = (V; E) with xed jV j and jEj is at least jEj? 3jV j log jV j log((jV j 2)=jEj) ? jV j ? 1. In other words, for most graphs, (H) = (jEj).
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تاریخ انتشار 2007