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The irregularity cost or sum of a graph. (English) Zbl 0780.05052

A multigraph is called irregular if no two of its vertices have the same degree. It is well known that no graph is irregular. However, if \(G\) is a graph having at most one isolated vertex and no component isomorphic to \(K_ 2\) then there is an irregular multigraph \(H\) containing \(G\) as its underlying graph. In F. Harary and O. R. Oellermann [The irregularity cost of a graph, to appear], the irregularity cost \(\text{ic}(G)\) of a graph \(G\) was defined to be the minimum number of edges that must be added to \(G\) to obtain an irregular multigraph (having \(G\) as its underlying graph). Independently, in [M. S. Jacobson, E. Kubicka and G. Kubicki] the irregularity sum \(s(G)\) was defined to be twice the minimum number of edges in a multigraph having \(G\) as its underlying graph. (The definition of irregularity sum given in this paper is in error due to an apparent typo.) Then clearly, \(s(G)=2(\text{ic}(G)+| E(G)|)\). Thus essentially the same concept was discovered independently. Furthermore, there are many common results in the two aforementioned papers. In this paper, many of the common results are presented without proof. Each team will publish their full papers separately.

MSC:

05C99 Graph theory
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References:

[1] F. Harary and O.R. Oellermann, The irregularity cost of a graph (to appear).; F. Harary and O.R. Oellermann, The irregularity cost of a graph (to appear). · Zbl 0901.05086
[2] M.S. Jacobson, E. Kubicka and G. Kubicki, Irregularity sum for graphs (to appear).; M.S. Jacobson, E. Kubicka and G. Kubicki, Irregularity sum for graphs (to appear). · Zbl 0947.05068
[3] Chartrand, G.; Lesniak, L., Graphs and Diagraphs (1986), Wadsworth: Wadsworth Monterey
[4] Harary, F., Graph Theory (1969), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0797.05064
[5] Chartrand, G.; Jacobson, M.; Lehel, J.; Oellermann, O.; Ruiz, S.; Saba, F., Irregular networks, Congressus Numerantium, 64, 197-210 (1988)
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