We review the class of continuous latent space (statistical) models for network data, paying particular attention to the role of the geometry of the latent space. In these models, the presence/absence of network dyadic ties are assumed to be conditionally independent given the dyads? unobserved positions in a latent space. In this way, these models provide a probabilistic framework for embeddin...

We study three mathematical notions, that of nodal regions for eigenfunctions of the Laplacian, that of covering theory, and that of fiber products, in the context of graph theory and spectral theory for graphs. We formulate analogous notions and theorems for graphs and their eigenpairs. These techniques suggest new ways of studying problems related to spectral theory of graphs. We also perform...

We review the Manhattan product of digraphs from the viewpoint of spectral analysis and obtain some preliminary formulae. As an example, the spectrum of the Manhattan product of the directed path Pn and the directed cycle C2 is obtained as well as its asymptotic spectral distribution.

In this comprehensive paper we have addressed structure-property relationships in a number of representative systems with periodic, random, quasi-periodic and deterministic aperiodic geometry using the interdisciplinary methods of spatial point pattern analysis and spectral graph theory as well as the rigorous Green’s matrix method, which provides access to the electromagnetic scattering behavi...

3 Graph Analysis 6 3.1 Analysis Based on Spectral Graph Theory . . . . . . . . . . . . . 7 3.2 Analysis Based on Random Field Theory . . . . . . . . . . . . . . 9 3.2.1 Markov Random Fields . . . . . . . . . . . . . . . . . . . . 9 3.2.2 Conditional Random Fields . . . . . . . . . . . . . . . . . 10 3.2.3 Gaussian Random Fields . . . . . . . . . . . . . . . . . . . 11 3.3 Analysis Based onMatri...

We investigate spectral properties of quantum graphs with infinitely many edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph with Kirchhoff or, more generally, δ-type couplings at vertices and the corresponding...

The median graph is a useful tool to cluster a set of graphs and obtain a prototype of them. The spectral graph theory is another approach to represent graphs and find “good” approximate solutions for the graph-matching problem. Recently, both approaches have been put together and a new representation has emerged, which is called Spectral-Median Graphs. In this paper, we summarize and compare t...

The proof of the theorem uses a SDP relaxation in terms of vectors vi ∈ Rn for all i ∈ V . Define distances to be d(i, j) ≡ ‖vi − vj‖ and balls to be B(i, r) ≡ {j ∈ V | d(i, j) ≤ r}. We first showed that if there exists a vertex i ∈ V such that |B(i, 1/4)| ≥ n/4, then we can find a cut of sparsity ≤ O(1) ·OPT . If there does not exist such a vertex in V , then we can find U ⊆ V with |U | ≥ n/2 ...

In this paper, we obtain some upper and lower bounds for the general extended energy of a graph. As an application, we obtain few bounds for the (edge) Zagreb energy of a graph. Also, we deduce a relation between Zagreb energy and edge-Zagreb energy of a graph $G$ with minimum degree $delta ge2$. A lower and upper bound for the spectral radius of the edge-Zagreb matrix is obtained. Finally, we ...