Let G be a simple graph on n vertices. LetH be either the complete graph Km or the complete bipartite graph Kr,s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if βi,α(H) ≤ βi,α(G) for all i ≥ 0 and α ∈ Z. In fact, it suffices to consider only the first syzygy module. In particular, we prove that β1,α(H) ≤ β1,α(G) for all α ∈ Z if and only if G contains a ...

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector-valued functions, called filtering functions. As is well known, in the case of scalar-valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, i.e. the dimensions of the images of the homomorphisms induced by the inclusions o...

Let G be a simple graph on n vertices and JG denote the binomial edge ideal of in polynomial ring S=K[x1,…,xn,y1,…,yn]. In this article, we compute second graded Betti numbers JG, obtain minimal presentation it when is tree or unicyclic graph. We classify all graphs whose ideals are almost complete intersection, prove that they generated by d-sequence Rees algebra their Cohen-Macaulay. also an ...

We prove that the Betti numbers of simplicial complexes of bounded vertex degrees are testable in constant time.

A generalization of the formula of Fine and Rao for the ranks of the intersection homology groups of a complex algebraic variety is given. The proof uses geometric properties of intersection homology and mixed Hodge theory. The middle-perversity intersection homology with integral coefficients of a compact complex n-dimensional algebraic variety X with isolated singularities is well known to be...