A unified construction of semiring-homomorphic graph invariants

Abstract

It has recently been observed by Zuiddam that finite graphs form a preordered commutative semiring under the graph homomorphism preorder together with join and disjunctive product as addition and multiplication, respectively. This led to a new characterization of the Shannon capacity \(\Theta \) via Strassen’s Positivstellensatz: \(\Theta ({\bar{G}}) = \inf _f f(G)\), where \(f: \mathsf {Graph}\rightarrow \mathbb {R}_+\) ranges over all monotone semiring homomorphisms. Constructing and classifying graph invariants \(\mathsf {Graph}\rightarrow \mathbb {R}_+\) which are monotone under graph homomorphisms, additive under join, and multiplicative under disjunctive product is therefore of major interest. We call such invariants semiring-homomorphic. The only known such invariants are all of a fractional nature: the fractional chromatic number, the projective rank, the fractional Haemers bounds, as well as the Lovász number (with the latter two evaluated on the complementary graph). Here, we provide a unified construction of these invariants based on linear-like semiring families of graphs. Along the way, we also investigate the additional algebraic structure on the semiring of graphs corresponding to fractionalization. Linear-like semiring families of graphs are a new notion of combinatorial geometry different from matroids which may be of independent interest.