2,489 research outputs found
Evaluating Matrix Functions by Resummations on Graphs: the Method of Path-Sums
We introduce the method of path-sums which is a tool for exactly evaluating a
function of a discrete matrix with possibly non-commuting entries, based on the
closed-form resummation of infinite families of terms in the corresponding
Taylor series. If the matrix is finite, our approach yields the exact result in
a finite number of steps. We achieve this by combining a mapping between matrix
powers and walks on a weighted directed graph with a universal graph-theoretic
result on the structure of such walks. We present path-sum expressions for a
matrix raised to a complex power, the matrix exponential, matrix inverse, and
matrix logarithm. We show that the quasideterminants of a matrix can be
naturally formulated in terms of a path-sum, and present examples of the
application of the path-sum method. We show that obtaining the inversion height
of a matrix inverse and of quasideterminants is an NP-complete problem.Comment: 23 pages, light version submitted to SIAM Journal on Matrix Analysis
and Applications (SIMAX). A separate paper with the graph theoretic results
is available at: arXiv:1202.5523v1. Results for matrices over division rings
will be published separately as wel
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
A sharp threshold for random graphs with a monochromatic triangle in every edge coloring
Let be the set of all finite graphs with the Ramsey property that
every coloring of the edges of by two colors yields a monochromatic
triangle. In this paper we establish a sharp threshold for random graphs with
this property. Let be the random graph on vertices with edge
probability . We prove that there exists a function with
, as tends to infinity
Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat
c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is
of independent interest is a generalization of Szemer\'edi's Regularity Lemma
to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.
- …