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Scaffolds: a graph-based system for computations in Bose-Mesner algebras

Let $X$ be a finite set and let $\mathsf{Mat}_X(\mathbb{C})$ denote the algebra of matrices with rows and columns indexed by $X$ and entries from the complex numbers acting on $\mathbb{C}^X$ with standard basis $\{ \hat{x} \mid x\in X\}$. For a digraph $G=(V(G),E(G))$, function $R:[m] \rightarrow V(G)$ with $r_j := R(j)$, and a function $w$ from the arcs of $G$ to $\mathsf{Mat}_X(\mathbb{C})$, we define the "scaffold" $\mathsf{S}(G,R;w)$ as the sum over all functions $φ$ from $V(G)$ to $X$ of the $m$-fold tensors $\widehat{φ(r_1)} \otimes \widehat{φ(r_2)} \otimes \cdots \otimes \widehat{φ(r_m)}$ scaled by the product of the entries $w(e)_{φ(a),φ(b)}$ over all arcs $e=(a,b)$ of $G$. Scaffolds can be used to count, among other things, digraph homomorphisms and association scheme parameters such as generalized intersection numbers. They also arise in the the theory of link invariants and spin models. These diagrams were introduced in the late 1980s by Arnold Neumaier and have been used implicitly by various authors working with association schemes. We revisit results of several authors, rephrasing their proofs in terms of these diagrams and certain rules of manipulation (or "moves") on diagrams. Our goal is to collect and present, in a uniform fashion, Neumaier's original idea extended to tensors and its used by various authors. Sometimes the term "star-triangle diagram" appears for what we, in this paper, call "scaffolds". Restricting to the case where edge weights are chosen from a coherent algebra, we explore the vector space $\mathsf{W}((G,R); \mathbb{A})$ spanned by all scaffolds defined on rooted diagram $(G,R)$ and establish a connection to graph minors. We end with a conjecture about planar scaffolds that draws a connection between association scheme duality and the duality of plane graphs.