The role of the normalized modularity matrix in finding homogeneous cuts will be presented. We also discuss the testability of the structural eigenvalues and that of the subspace spanned by the corresponding eigenvectors of this matrix. In the presence of a spectral gap between the k− 1 largest absolute value eigenvalues and the remainder of the spectrum, this in turn implies the testability of...

The energy of a graph was introduced by Gutman in 1978 as the sum of the absolute values of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. These are Cayley graphs on cyclic groups (i.e. there adjacency matrix is circulant) each of whose eigenvalues is an integer. Given an arbitrary prime power p, we determine all integral circu...

We present a minimization principle for the sum of the first few smallest positive eigenvalues and Cauchy-like interlacing inequalities for the linear response eigenvalue problem arising from the calculation of excitation states of many-particle systems, a hot topic among computational material scientists today for materials design to advance energy science. Subsequently, we develop the best ap...

Abstract We use the method of similar operators to study a mixed problem for first order differential equation with fractional integration operator. The operator defined by is transformed into that an orthogonal direct sum finite-rank operators. estimates eigenvalues, eigenvectors are obtained. result used construct group.

Alfred Horn's conjecture on eigenvalues of sums Hermitian matrices was proved more than 20 years ago. In this note, the problem is raised of, given an $n$-tuple $\gamma$ in solution polytope, constructing with required spectra such that their sum has $\gamma$.

The eccentricity e(v) of a vertex v in a connected graph G is the distance between v and a vertex furthest from v in G. The center C(G) of G is the subgraph induced by those vertices of G having minimum eccentricity; the periphery P(G) is the subgraph induced by those vertices of G having maximum eccentricity. The distance d(v) of a vertex v in G is the sum of the distances from v to the vertic...

Let A(G) denote the adjacency matrix of the graph G. The polynomial pA(G)(x) is usually referred to as the characteristic polynomial of G. For convenience, we use p(G, x) to denote pA(G)(x). The spectrum of a graph G is the set of eigenvalues of A(G) together with their multiplicities. Since A (short for A(G)) is a real symmetric matrix, basic linear algebra tells us a few thing about A and its...

Let T be a linear compact operator on a Hilbert space H, Aj be its eigenvalues, IA,1 > /& > ,..., rj be the moduli of the real parts of the eigenvalues ordered so that r, > r2 .... Note that rj is not necessarily equal to 1 Re ;ljj. Let Lj be the eigensubspace of T corresponding to Aj, yj be the eigensubspace of T corresponding to rj, zj = Cjk,, i L,, aj = xi=, -k Mk. Let tj be the moduli of th...

In this paper, some inequality relations between the Laplacian/signless Laplacian H-eigenvalues and the clique/coclique numbers of uniform hypergraphs are presented. For a connected uniform hypergraph, some tight lower bounds on the largest Laplacian H+-eigenvalue and signless Laplacian H-eigenvalue related to the clique/coclique numbers are given. And some upper and lower bounds on the clique/...