One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenva...

In the present paper MAPLE subroutines computing Milnor and Tyurina numbers of an isolated algebraic hypersurface singularity are presented and described in detail. They represents examples, and perhaps the first ones, of a MAPLE implementation of local monomial ordering. As an application, the last section is devoted to writing down equations of algebraic stratifications of Kuranishi spaces of...

The weighted Sitting Closer to Friends than Enemies (SCFE) problem is find an injection of the vertex set a given graph into metric space so that, for every pair incident edges with different weight, end vertices heavier edge are closer lighter edge. Seriation simultaneous reordering rows and columns symmetric matrix such that entries monotone nondecreasing in when moving towards diagonal. If e...

Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in precisely one of the bicliques in the ...

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or reducible. As related objects, we study vertex sets with maximum cut (edge boundary) in D(m,n) and prove some facts on their structure. We show that the considered tw...

Submodularity is an important  property of set functions with deep theoretical results  and various  applications. Submodular systems appear in many applicable area, for example machine learning, economics, computer vision, social science, game theory and combinatorial optimization.  Nowadays submodular functions optimization has been attracted by many researchers.  Pendant pairs of a symmetric...

We describe a statistical approach for modeling agreements and disagreements in conversational interaction. Our approach first identifies adjacency pairs using maximum entropy ranking based on a set of lexical, durational, and structural features that look both forward and backward in the discourse. We then classify utterances as agreement or disagreement using these adjacency pairs and feature...