Article

Ø(G) V. Y. Dobrynin

January 1998

Authors:

Vladimir Dobrynin

Saint Petersburg State University

A new function of a graph G is presented. Say that a matrix B that is indexed by vertices of G is feasible for G if it is real, symmetric and I B I +A(G); where I is the identity matrix and A(G) is the adjacency matrix of G. Let B(G) be the set of all feasible matrices for G, and let Ø(G) be the smallest number of cliques that cover the vertices of G. We show that ff(G) minf rank(B) jB 2 B(G)g Ø(G) and that ff(G) = minf rank(B) jB 2 B(G)g implies ff(G) = Ø(G): The well known Lov'asz number #(G) of a graph G [1] is "sandwiched" between the size of the largest stable set in G and the smallest number of cliques that cover the vertices of G ff(G) #(G) Ø(G): Some alternative definitions of #(G) are introduced in [2,3]. For example, #(G) = maxf (B) jB is a real positive semidefinite matrix indexed by vertices of G, B vv = 1 for all v 2 V (G), B uv = 0 whenever u GammaGamma v in Gg; where (B) is the maximum eigenvalue of B, V (G) --- the set of vertices of G, u GammaGamma v ...

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