120 research outputs found
Some results on the saturation number for unions of cliques
Graph is -saturated if is not a subgraph of and is a
subgraph of for any edge not in . The saturation number for a
graph is the minimal number of edges in any -saturated graph of order
. In this paper, the saturation number for ( and ) is determined, and the extremal graph for is determined. Moreover, the saturation number and the extremal graph for
() are completely determined
Linear algebra and bootstrap percolation
In \HH-bootstrap percolation, a set A \subset V(\HH) of initially
'infected' vertices spreads by infecting vertices which are the only uninfected
vertex in an edge of the hypergraph \HH. A particular case of this is the
-bootstrap process, in which \HH encodes copies of in a graph . We
find the minimum size of a set that leads to complete infection when
and are powers of complete graphs and \HH encodes induced copies of
in . The proof uses linear algebra, a technique that is new in bootstrap
percolation, although standard in the study of weakly saturated graphs, which
are equivalent to (edge) -bootstrap percolation on a complete graph.Comment: 10 page
Induced Saturation Number
In this paper, we discuss a generalization of the notion of saturation in
graphs in order to deal with induced structures. In particular, we define , which is the fewest number of gray edges in a trigraph so that
no realization of that trigraph has an induced copy of , but changing any
white or black edge to gray results in some realization that does have an
induced copy of .
We give some general and basic results and then prove that for where is the path on 4
vertices. We also show how induced saturation in this setting extends to a
natural notion of saturation in the context of general Boolean formulas.Comment: 14 pages, 7 figure
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Combinatorics
Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combiantorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session
Core–satellite graphs : clustering, assortativity and spectral properties
Core-satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts-Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In addition, we completely describe the spectrum of the adjacency and Laplacian matrices associated with core-satellite graphs. Finally, we introduce the class of generalized core-satellite graphs and analyze their clustering, assortativity, and spectral properties
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