Graphs admitting only constant splines (Q2305496)

Splines appear in a surprising number of different guises throughout the literature on pure and applied mathematics. On the one hand, splines are a classical topic in numerical analysis [\textit{M.-J. Lai} and \textit{L. L. Schumaker}, Spline functions on triangulations. Cambridge: Cambridge University Press (2007; Zbl 1185.41001) ] and computer-aided geometric design [\textit{G. Farin}, Curves and surfaces for computer aided geometric design. A practical guide. Boston etc.: Academic Press (1990; Zbl 0702.68004)]. On the other hand, splines appear in a quite different way as equivariant cohomology rings of topological spaces with a torus action via GKM theory [\textit{M. Goresky} et al., Invent. Math. 131, No. 1, 25--83 (1998; Zbl 0897.22009)]. A nice (and accessible) survey of this latter perspective appears in [\textit{J. Tymoczko}, Comput. Aided Geom. Des. 45, 32--47 (2016; Zbl 1418.41013)]. Generalized splines on graphs, introduced in [\textit{S. Gilbert} et al., Pac. J. Math. 281, No. 2, 333--364 (2016; Zbl 1331.05183)], provide a unifying perspective for splines arising in different contexts. More precisely, suppose we are given a graph \(G=(V,E)\), a commutative ring \(R\) with unit, and a map \(\alpha: E\to I(R)\), where \(I(R)\) is the set of ideals of \(R\) (\(\alpha\) is called an \textit{edge labeling}). A \textit{spline} on the edge-labeled graph \((G,\alpha)\) is a vertex labeling \(p:V\to R\) satisfying that if \(e=\{u,v\}\) is an edge of \(G\), then \(p(u)-p(v)\in\alpha(e)\). By an appropriate choice of ring, graph, and edge-labeling, one may recover classical splines or the equivariant cohomology rings of GKM theory. In the paper under review, the authors focus on the ring structure of \(S_R(G,\alpha)\). The main result, inspiring the title of the paper, is a combinatorial characterization in Theorem~3.5 of those edge-labeled graphs which only admit constant splines (a \textit{constant spline} \(p:V\to R\) takes the same value at every vertex). The characterization works for rings of the form \(R\cong \bigoplus_{i=1}^k R_i\), where \(R_1,\ldots,R_k\) are irreducible (a ring is \textit{irreducible} if the intersection of any two non-zero ideals is non-zero). The result is reminiscent of sparsity conditions in rigidity [\textit{A. Frank} and \textit{L. Szegő}, Discrete Appl. Math. 131, No. 2, 347--371 (2003; Zbl 1022.05071)].