The Clique Numbers of Regular Graphs

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.