The Arithmetic Tutte polynomial of two matrices associated to Trees

Abstract

Abstract Arithmetic matroids arising from a list A of integral vectors in Zn are of recent interest and the arithmetic Tutte polynomial MA(x, y) of A is a fundamental invariant with deep connections to several areas. In this work, we consider two lists of vectors coming from the rows of matrices associated to a tree T. Let T = (V, E) be a tree with |V| = n and let LT be the q-analogue of its Laplacian L in the variable q. Assign q = r for r ∈ ℤ with r/= 0, ±1 and treat the n rows of LT after this assignment as a list containing elements of ℤn. We give a formula for the arithmetic Tutte polynomial MLT (x, y) of this list and show that it depends only on n, r and is independent of the structure of T. An analogous result holds for another polynomial matrix associated to T: EDT, the n × n exponential distance matrix of T. More generally, we give formulae for the multivariate arithmetic Tutte polynomials associated to the list of row vectors of these two matriceswhich shows that even the multivariate arithmetic Tutte polynomial is independent of the tree T. As a corollary, we get the Ehrhart polynomials of the following zonotopes: - ZEDT obtained from the rows of EDT and - ZLT obtained from the rows of LT. Further, we explicitly find the maximum volume ellipsoid contained in the zonotopes ZEDT, ZLT and show that the volume of these ellipsoids are again tree independent for fixed n, q. A similar result holds for the minimum volume ellipsoid containing these zonotopes.