Haemers Number

The Haemers number of an n-vertex graph G, denoted H(G), H(G) (Alipour abd Gohari 2023), or R(G) (Haemers 1978), is an integer defined as the minimum rank over all n×n matrices B over some field such that b_(ii)!=0 and b_(ij)=0 if vertices i and j are not adjacent in a given graph G. (Note that the critical word "not" was inadvertently omitted in the original Haemers (1978) paper.)

The Haemers number provodes upper bound on the Shannon capacity of G which is sometimes better than the Lovász number.

The Haemers number satisfies


(Haemers 1978), where chi is the chromatic number and G^_ denotes the graph complement of G.

See also

Lovász Number, Shannon Capacity

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Alipour, S. and Gohari, A. "Relative Fractional Independence Number and Its Applications." 17 Jul 2023., W. H. "An Upper Bound for the Shannon Capacity of a Graph." Colloq. Math. Soc. János Bolyai 25, 267-272, 1978.Taziki, M. "Relative Fractional Packing Number and Its Properties." 28 Nov 2023.

Cite this as:

Weisstein, Eric W. "Haemers Number." From MathWorld--A Wolfram Web Resource.