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The sum number of a disjoint union of graphs
TLDR
In this paper, the authors considered the disjoint union of graphs as sum graphs and provided an upper bound on the sum number of such graphs and an application for the exclusive sum number.Abstract:
In this paper we consider the disjoint union of graphs as sum graphs. We provide an upper bound on the sum number of a disjoint union of graphs and provide an application for the exclusive sum number of a graph. We conclude with some open problems.read more
Citations
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Journal ArticleDOI
Minimal sum labeling of graphs
TL;DR: If the conditions are relaxed (either allow non-injective labelings or consider graphs with loops) then there are sum graphs without a minimal labeling, which partially answers the question posed by Miller, Ryan and Smyth.
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Sum Labelling Graphs of Maximum Degree Two
Henning Fernau,Kshitij Gajjar +1 more
TL;DR: In this paper , it was shown that for graphs of maximum degree two, the disjoint union of paths and cycles is the union of all the vertices in the graph.
References
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Journal Article
Labelling wheels for minimum sum number
TL;DR: For even n 4 and odd n 5, it was shown in this paper that oe(W n ) = n 2 + 2 for wheels W n of order n+1 and size m = 2n, where m is the size of the graph.
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A family of sparse graphs of large sum number
Nora Hartsfield,William F. Smyth +1 more
TL;DR: In this paper, it was shown that for wheels Wn of (sufficiently large) order n + 1 and size m = 2n, σ(Wn) = n/2 + 3 if n is even and n ⩽ σ (Wn), ⌈ n + 2 if n was odd.
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New formula for the sum number for the complete bipartite graphs
TL;DR: The sum number σ(G) of the graph G is the least number of isolated vertices one must add to G to turn it into a sum graph and the new construction given in this paper shows that σ (K m,n ) in this case is much smaller.
The sum number of the cocktail party graph
TL;DR: In this paper, the complete n-partite graph G = H m;n was considered and an optimal labeling of the vertices of G by distinct positive integers was given, where the vertex u and v are adjacent if and only if there exists a vertex u + v. This was the first known graph with this property.