65,865 research outputs found
Sandwiching saturation number of fullerene graphs
The saturation number of a graph is the cardinality of any smallest
maximal matching of , and it is denoted by . Fullerene graphs are
cubic planar graphs with exactly twelve 5-faces; all the other faces are
hexagons. They are used to capture the structure of carbon molecules. Here we
show that the saturation number of fullerenes on vertices is essentially
Pattern matching and pattern discovery algorithms for protein topologies
We describe algorithms for pattern matching and pattern
learning in TOPS diagrams (formal descriptions of protein topologies).
These problems can be reduced to checking for subgraph isomorphism
and finding maximal common subgraphs in a restricted class of ordered
graphs. We have developed a subgraph isomorphism algorithm for
ordered graphs, which performs well on the given set of data. The
maximal common subgraph problem then is solved by repeated
subgraph extension and checking for isomorphisms. Despite the
apparent inefficiency such approach gives an algorithm with time
complexity proportional to the number of graphs in the input set and is
still practical on the given set of data. As a result we obtain fast
methods which can be used for building a database of protein
topological motifs, and for the comparison of a given protein of known
secondary structure against a motif database
On neighbour sum-distinguishing -edge-weightings of bipartite graphs
Let be a set of integers. A graph G is said to have the S-property if
there exists an S-edge-weighting such that any two
adjacent vertices have different sums of incident edge-weights. In this paper
we characterise all bridgeless bipartite graphs and all trees without the
-property. In particular this problem belongs to P for these graphs
while it is NP-complete for all graphs.Comment: Journal versio
Quantum state-independent contextuality requires 13 rays
We show that, regardless of the dimension of the Hilbert space, there exists
no set of rays revealing state-independent contextuality with less than 13
rays. This implies that the set proposed by Yu and Oh in dimension three [Phys.
Rev. Lett. 108, 030402 (2012)] is actually the minimal set in quantum theory.
This contrasts with the case of Kochen-Specker sets, where the smallest set
occurs in dimension four.Comment: 8 pages, 2 tables, 1 figure, v2: minor change
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