Spectra of infinite graphs via Schur complement

A Survey on Spectra of Infinite Graphs

In the last decade, considerable attention has been paid by graph theorists to the study of spectra of graphs and their interaction with structure and characteristic properties of graphs. After the 'classical' book by N. L. Biggs [12] on algebraic graph theory in general, the first comprehensive monograph on this particular topic, by D. M. Cvetkovic, M. Doob and H. Sachs [34], appeared in 1979....

Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs

We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that contain small balanced separators. We give a fully dynamic algorithm that maintains (1 + ε)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with Õ( √ n/ε) worst-case update time and Õ( √ n...

On Preconditioning Schur Complement and Schur Complement Preconditioning

We study two implementation strategies to utilize Schur complement technique in multilevel recursive incomplete LU preconditioning techniques (RILUM) for solving general sparse matrices. The first strategy constructs a RILUM to precondition the original matrix. The second strategy solves the first Schur complement matrix using the lower level parts of the RILUM as the preconditioner. We discuss...

INTEGER-MAGIC SPECTRA OF CYCLE RELATED GRAPHS

For any h in N , a graph G = (V, E) is said to be h-magic if there exists a labeling l: E(G) to Z_{h}-{0} such that the induced vertex set labeling l^{+: V(G) to Z_{h}} defined by l^{+}(v)= Summation of l(uv)such that e=uvin in E(G) is a constant map. For a given graph G, the set of all for which G is h-magic is called the integer-magic spectrum of G and is denoted by IM(G). In this paper, the ...

Applications of Dual Schur Complement