Toric multivariate Gaussian models from symmetries in a tree

Given a rooted tree T on n non-root leaves with colored and zeroed nodes, we construct a linear space L T of n × n symmetric matrices with constraints determined by the combinatorics of the tree. When L T represents the covariance matrices of a Gaussian model, it provides natural generalizations of Brownian motion tree (BMT) models in phylogenetics and a step toward a more accurate model for phylogenetic networks with symmetries for species hybridization. When L T represents a space of concentration matrices of a Gaussian model, it gives certain colored Gaussian graphical models, which we refer to as BMT derived models. We investigate conditions under which the reciprocal variety L T − 1 is toric. Relying on the birational isomorphism of the inverse matrix map, we show that if the BMT derived graph of T is vertex-regular and a block graph, under the derived Laplacian transformation, which we introduce, L T − 1 is the vanishing locus of a toric ideal. This ideal is given by the sum of the toric ideal of the Gaussian graphical model on the block graph, the toric ideal of the original BMT model, and binomial linear conditions coming from vertex-regularity. To this end, we provide monomial parametrizations for these toric models realized through paths among leaves in T . The reciprocal variety L T − 1 is toric under the G -derived Laplacian, with toric vanishing ideal 〈 p 14 − p 24 , p 13 − p 23 , p 01 − p 02 , p 03 p 24 − p 02 p 34 , p 04 p 23 − p 24 p 34 〉 . • Brownian Motion Tree (BMT) models are generalized via added covariance symmetries. • Generalized BMT models connect to colored Gaussian graphical models. • Generalized BMT models can represent phylogenetic networks with species hybridization. • Combinatorial conditions on the tree guarantee the model is a toric variety. • We give monomial parametrizations and binomial invariants of these toric models.