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.

Zuiddam最近发现，有限图在图同态前序下与连接和析取积分别作为加法和乘法一起形成了一个有序交换半环。这导致通过Strassen的Positivstellensatz对香农容量\（\ Theta \）进行了新的表征：\（\ Theta（{\ bar {G}}）= \ inf _f f（G）\），其中\（f：\ mathsf {Graph} \ rightarrow \ mathbb {R} _ + \）覆盖所有单调半环同构。因此，构造和分类图不变性\（\ mathsf {Graph} \ rightarrow \ mathbb {R} _ + \）是图同态下的单调，联接下的加法和析取积下的可乘的。我们称这样的不变量半环同态的。唯一已知的此类不变量具有分数性质：分数色数，投影秩，分数Haemers界以及Lovász数（后两个在互补图中求值）。在这里，我们基于图的线性半环族提供了这些不变量的统一构造。在此过程中，我们还研究了图的半环上对应于分数化的其他代数结构。线性半环图族是组合几何学的新概念，不同于拟阵可能是独立引起关注的。