59,576 research outputs found
On the Signed -independence Number of Graphs
In this paper, we study the signed 2-independence number in graphs and give new sharp upper and lower bounds on the signed 2-independence number of a graph by a simple uniform approach. In this way, we can improve and generalize some known results in this area
Bicircular signed-graphic matroids
Several matroids can be defined on the edge set of a graph. Although
historically the cycle matroid has been the most studied, in recent times, the
bicircular matroid has cropped up in several places. A theorem of Matthews from
late 1970s gives a characterization of graphs whose bicircular matroids are
graphic. We give a characterization of graphs whose bicircular matroids are
signed-graphic.Comment: 8 page
A semidefinite programming hierarchy for packing problems in discrete geometry
Packing problems in discrete geometry can be modeled as finding independent
sets in infinite graphs where one is interested in independent sets which are
as large as possible. For finite graphs one popular way to compute upper bounds
for the maximal size of an independent set is to use Lasserre's semidefinite
programming hierarchy. We generalize this approach to infinite graphs. For this
we introduce topological packing graphs as an abstraction for infinite graphs
coming from packing problems in discrete geometry. We show that our hierarchy
converges to the independence number.Comment: (v2) 25 pages, revision based on suggestions by referee, accepted in
Mathematical Programming Series B special issue on polynomial optimizatio
A bivariate chromatic polynomial for signed graphs
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial
which counts all -colorings of a graph such
that adjacent vertices get different colors if they are . Our first
contribution is an extension of to signed graphs, for which we
obtain an inclusion--exclusion formula and several special evaluations giving
rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is
to derive combinatorial reciprocity theorems for and its
signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking
chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure
Emergent Behaviors over Signed Random Networks in Dynamical Environments
We study asymptotic dynamical patterns that emerge among a set of nodes that
interact in a dynamically evolving signed random network. Node interactions
take place at random on a sequence of deterministic signed graphs. Each node
receives positive or negative recommendations from its neighbors depending on
the sign of the interaction arcs, and updates its state accordingly. Positive
recommendations follow the standard consensus update while two types of
negative recommendations, each modeling a different type of antagonistic or
malicious interaction, are considered. Nodes may weigh positive and negative
recommendations differently, and random processes are introduced to model the
time-varying attention that nodes pay to the positive and negative
recommendations. Various conditions for almost sure convergence, divergence,
and clustering of the node states are established. Some fundamental
similarities and differences are established for the two notions of negative
recommendations
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