The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph. The structure of this group is a subtle isomorphism invariant that has received much attention recently, partly due to its relation to the graph Laplacian and chip-firing games. However, the group structure has been determined for relatively few classes of graphs. We conj...

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. S...

We propose a graph spectrum-based Gaussian process for prediction of signals defined on nodes the graph. The model is designed to capture various signal structures through highly adaptive kernel that incorporates flexible polynomial function in spectral domain. Unlike most existing approaches, we learn such kernel, where setup enables learning without need eigen-decomposition Laplacian. In addi...

This paper proposes a novel nonparametric framework for semi-supervised learning and for optimizing the Laplacian spectrum of the data manifold simultaneously. Our formulation leads to a convex optimization problem that can be efficiently solved via the bundle method, and can be interpreted as to asymptotically minimize the generalization error bound of semi-supervised learning with respect to ...

In this paper, we investigate the properties of the cell-graphs by using the spectral graph theory. The spectral analysis is performed on (i) the adjacency matrix of a cell-graph, and (ii) the normalized Laplacian of the cell-graph. We show that the spectra of the cell-graphs of cancerous tissues are unique and the features extracted from these spectra distinguish the cancerous (malignant gliom...

let $a = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrixwhere $n geq 2$. let $dt(a)$, its second immanant be the immanant corresponding to the partition $lambda_2 = 2,1^{n-2}$. let $g$ be a connected graph with blocks $b_1, b_2, ldots b_p$ and with$q$-exponential distance matrix $ed_g$. we given an explicitformula for $dt(ed_g)$ which shows that $dt(ed_g)$ is independent of the manner in ...

A graph G is said to be determined by the spectrum of its Laplacian matrix (DLS) if every with same isomorphic G. In some recent papers it proved that friendship graphs and starlike trees are DLS. If a tree joined merging their vertices degree greater than two, then resulting called path-friendship graph. this paper, graphs, natural generalization trees, also Consequently, using these results w...

How does coarsening affect the spectrum of a general graph? We provide conditions such that the principal eigenvalues and eigenspaces of a coarsened and original graph Laplacian matrices are close. The achieved approximation is shown to depend on standard graph-theoretic properties, such as the degree and eigenvalue distributions, as well as on the ratio between the coarsened and actual graph s...

Grone and Merris [5] conjectured that the Laplacian spectrum of a graph is majorized by its conjugate vertex degree sequence. In this paper, we prove that this conjecture holds for a class of graphs including trees. We also show that this conjecture and its generalization to graphs with Dirichlet boundary conditions are equivalent.