For an induced subgraph S of a graph, we show that its Neumann eigenvalue λS can be lower-bounded by using the heat kernel Ht(x, y) of the subgraph. Namely, λS ≥ 1 2t ∑ x∈S inf y∈S Ht(x, y) √ dx √ dy where dx denotes the degree of the vertex x. In particular, we derive lower bounds of eigenvalues for convex subgraphs which consist of lattice points in an d-dimensional Riemannian manifolds M wit...

We give a simple proof of generalization an inequality for homomorphism counts by Sidorenko (1994). A special case our says that if $d_v$ denotes the degree vertex $v$ in graph $G$ and $\textrm{Hom}_\Delta(H,G)$ number homomorphisms from connected $H$ on $h$ vertices to which map particular with $d_v \ge \Delta$, then $\textrm{Hom}_\Delta(H,G) \le \sum_{v\in G} d_v^{h-1}\mathbf{1}_{d_v\ge \Delt...

In this article we provide hardness results and approximation algorithms for the following three natural degree-constrained subgraph problems, which take as input an undirected graph G = (V, E). Let d ≥ 2 be a fixed integer. The Maximum d-degree-bounded Connected Subgraph (MDBCSd) problem takes as additional input a weight function ω : E → R, and asks for a subset E′ ⊆ E such that the subgraph ...

A bipartite graph G = (L,R;E) with at least one edge is said to be identifiable if for every vertex v ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L| − 1. An `-subgraph of G is an induced subgraph of G obtained by deleting from it some vertices in L together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given...