Published September 14, 2021
| Version v1
Publication
Gradation in Greyscales of Graphs
Description
In this work we present the notion of greyscale of a graph as a colouring
of its vertices that uses colours from the real interval [0,1]. Any greyscale
induces another colouring by assigning to each edge the non-negative dif-
ference between the colours of its vertices. These edge colours are ordered
in lexicographical decreasing ordering and give rise to a new element of the
graph: the gradation vector. We introduce the notion of minimum grada-
tion vector as a new invariant for the graph and give polynomial algorithms
to obtain it. These algorithms also output all greyscales that produce the
minimum gradation vector. This way we tackle and solve a novel vectorial
optimization problem in graphs that may produce more satisfactory solu-
tions than those ones generated by known scalar optimization approaches.
The interest of these new concepts lies in their possible applications for
solving problems of engineering, physics and applied mathematics which
are modeled according to a network whose nodes have assigned numerical
values of a certain parameter delimited by a range of real numbers. The ob-
jective is to minimize the differences between each node and its neighbors,
ensuring that the extreme values of the interval are assigned.
Abstract
Ministerio de Economía, Industria y Competitividad MTM2015-65397-PAbstract
Junta de Andalucía PAI FQM-164Additional details
Identifiers
- URL
- https://idus.us.es/handle//11441/125728
- URN
- urn:oai:idus.us.es:11441/125728