Abstract.
Let ω(G) be the clique number of a graph G. We prove that if G runs over the set of graphs with a fixed degree sequence d, then the values ω(G) completely cover a line segment [a,b] of positive integers. For an arbitrary graphic degree sequence d, we define min(ω,d) and max(ω,d) as follows:
where is the graph of realizations of d.
Thus the two invariants a:=min(ω,d) and b:=max(ω,d) naturally arise. For a graphic degree sequence d=r n:=(r,r,…,r) where r is the vertex degree and n is the number of vertices, the exact values of a and b are found in all situations. Since the independence number, α(G)=ω(G¯), we obtain parallel results for the independence number of graphs.
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Received: October, 2001 Final version received: July 25, 2002
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ID="*" Work supported by The Thailand Research Fund, under the grant number BRG/09/2545
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Punnim, N. The Clique Numbers of Regular Graphs. Graphs Comb 18, 781–785 (2002). https://doi.org/10.1007/s003730200064
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DOI: https://doi.org/10.1007/s003730200064