1,580 research outputs found
On the Computational Complexity of Defining Sets
Suppose we have a family of sets. For every , a
set is a {\sf defining set} for if is the
only element of that contains as a subset. This concept has been
studied in numerous cases, such as vertex colorings, perfect matchings,
dominating sets, block designs, geodetics, orientations, and Latin squares.
In this paper, first, we propose the concept of a defining set of a logical
formula, and we prove that the computational complexity of such a problem is
-complete.
We also show that the computational complexity of the following problem about
the defining set of vertex colorings of graphs is -complete:
{\sc Instance:} A graph with a vertex coloring and an integer .
{\sc Question:} If be the set of all -colorings of
, then does have a defining set of size at most ?
Moreover, we study the computational complexity of some other variants of
this problem
Graphs, Matrices, and the GraphBLAS: Seven Good Reasons
The analysis of graphs has become increasingly important to a wide range of
applications. Graph analysis presents a number of unique challenges in the
areas of (1) software complexity, (2) data complexity, (3) security, (4)
mathematical complexity, (5) theoretical analysis, (6) serial performance, and
(7) parallel performance. Implementing graph algorithms using matrix-based
approaches provides a number of promising solutions to these challenges. The
GraphBLAS standard (istc- bigdata.org/GraphBlas) is being developed to bring
the potential of matrix based graph algorithms to the broadest possible
audience. The GraphBLAS mathematically defines a core set of matrix-based graph
operations that can be used to implement a wide class of graph algorithms in a
wide range of programming environments. This paper provides an introduction to
the GraphBLAS and describes how the GraphBLAS can be used to address many of
the challenges associated with analysis of graphs.Comment: 10 pages; International Conference on Computational Science workshop
on the Applications of Matrix Computational Methods in the Analysis of Modern
Dat
GTI-space : the space of generalized topological indices
A new extension of the generalized topological indices (GTI) approach is carried out torepresent 'simple' and 'composite' topological indices (TIs) in an unified way. Thisapproach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randićconnectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index andreverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given
Fixing number of co-noraml product of graphs
An automorphism of a graph is a bijective mapping from the vertex set of
to itself which preserves the adjacency and the non-adjacency relations of
the vertices of . A fixing set of a graph is a set of those vertices
of which when assigned distinct labels removes all the automorphisms of
, except the trivial one. The fixing number of a graph , denoted by
, is the smallest cardinality of a fixing set of . The co-normal
product of two graphs and , is a graph having the
vertex set and two distinct vertices are adjacent if is adjacent to
in or is adjacent to in . We define a general
co-normal product of graphs which is a natural generalization of the
co-normal product of two graphs. In this paper, we discuss automorphisms of the
co-normal product of graphs using the automorphisms of its factors and prove
results on the cardinality of the automorphism group of the co-normal product
of graphs. We prove that , for
any two graphs and . We also compute the fixing number of the
co-normal product of some families of graphs.Comment: 13 page
Constructing Adjacency Arrays from Incidence Arrays
Graph construction, a fundamental operation in a data processing pipeline, is
typically done by multiplying the incidence array representations of a graph,
and , to produce an adjacency
array of the graph, , that can be processed with a variety of
algorithms. This paper provides the mathematical criteria to determine if the
product
will have the required structure of the adjacency array of the graph. The
values in the resulting adjacency array are determined by the corresponding
addition and multiplication operations used to perform the
array multiplication. Illustrations of the various results possible from
different and operations are provided using a small
collection of popular music metadata.Comment: 8 pages, 5 figures, accepted to IEEE IPDPS 2017 Workshop on Graph
Algorithm Building Block
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