We introduce the notions of a path complex and its homologies. Particular cases of path homologies are simplicial homologies and digraph homologies. We state and prove some properties of path homologies, in particular, the K¨unneth formulas for Cartesian product and join, which happen to be true at the level of chain complexes.

We prove the equivalence between some functional inequalities and the ultracontractivity property of the heat semigroup on infinite graphs. These functional inequalities include Sobolev inequalities, Nash inequalities, Faber–Krahn inequalities, and log-Sobolev inequalities. We also show that, under the assumptions of volume growth and CDE(n, 0), which is regarded as the natural notion of curvature on graphs, these four functional inequalities and the ultracontractivity property of the heat semigroup are all true on graphs.

Let G=(V,E) be a locally finite connected weighted graph, and Δ be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation u_t = Δu + f(u) on G. The blow-up phenomenons for u_t = Δu + f(u) are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

Based on a global estimate of the heat kernel, some important inequalities such as Poincaré inequality and log-Sobolev inequality, furthermore a tight logarithm Sobolev inequality are presented on graphs, just under a positive curvature condition CDE'(n,K) with some K > 0. As consequences, we obtain exponential integrability of integrable Lipschitz functions and moment bounds at the same assumption on graphs.