We introduce the class of lattice-linear monomial ideals and use the lcm-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with a linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets η : P → N n a sequence of multigraded modules and maps.

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the minimal free resolution of a larger class of rigid monomial ideals by using L(n), the lattice of all lcm-lattices of monomial ideals with n generators. By fixing a stratum in L(n) where all ideals have the same total Betti numbers we show that rigidity is a property which is upward closed in L(n). Furthermore, the minimal resolution of all rigid ideals contained in a fixed stratum is shown to be isomorphic to the constructed minimal resolution.

We use the theory of poset resolutions to give an alternate construction for the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.

We investigate the first and second moments of the inverse participation ratio (IPR) for all eigenvectors of the Laplacian on finite random regular graphs with n vertices and degree z. By exactly diagonalizing a large set of z-regular graphs, we find that as n becomes large, the mean of the inverse participation ratio on each graph, when averaged over a large ensemble of graphs, approaches the numerical value 3. This universal number is understood as the large-n limit of the average of the quartic polynomial corresponding to the IPR over an appropriate (n − 2)-dimensional hypersphere of R n . For a large, but not exhaustive ensemble of graphs, the mean variance of the inverse participation ratio for all graph Laplacian eigenvectors deviates from its continuous hypersphere average due to large graph-to-graph fluctuations that arise from the existence of highly localized modes.

We introduce the notion of a resolution supported on a poset. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution. Work of Velasco has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is instead a homology CW-poset that supports the minimal free resolution of the ideal. In general there is more than one choice for the isomorphism class of such a poset, and it is an open question whether there is a canonical one.