2,228 research outputs found
Ricci-flat graphs with girth at least five
A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges.
Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau,
Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009].
In this paper, we classified all Ricci-flat connected graphs with girth at
least five: they are the infinite path, cycle (), the
dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We
also construct many Ricci-flat graphs with girth 3 or 4 by using the root
systems of simple Lie algebras.Comment: 14 pages, 15 figure
Edge Roman domination on graphs
An edge Roman dominating function of a graph is a function satisfying the condition that every edge with
is adjacent to some edge with . The edge Roman
domination number of , denoted by , is the minimum weight
of an edge Roman dominating function of .
This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad stating that if is a graph of maximum degree
on vertices, then . While the counterexamples having the edge Roman domination numbers
, we prove that is an upper bound for connected graphs. Furthermore, we
provide an upper bound for the edge Roman domination number of -degenerate
graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi
and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic
graphs.
In addition, we prove that the edge Roman domination numbers of planar graphs
on vertices is at most , which confirms a conjecture of
Akbari and Qajar. We also show an upper bound for graphs of girth at least five
that is 2-cell embeddable in surfaces of small genus. Finally, we prove an
upper bound for graphs that do not contain as a subdivision, which
generalizes a result of Akbari and Qajar on outerplanar graphs
Excluded minors in cubic graphs
Let G be a cubic graph, with girth at least five, such that for every
partition X,Y of its vertex set with |X|,|Y|>6 there are at least six edges
between X and Y. We prove that if there is no homeomorphic embedding of the
Petersen graph in G, and G is not one particular 20-vertex graph, then either
G\v is planar for some vertex v, or G can be drawn with crossings in the plane,
but with only two crossings, both on the infinite region. We also prove several
other theorems of the same kind.Comment: 62 pages, 17 figure
Modeling and synthesis of multicomputer interconnection networks
The type of interconnection network employed has a profound effect on the performance of a multicomputer and multiprocessor design. Adequate models are needed to aid in the design and development of interconnection networks. A novel modeling approach using statistical and optimization techniques is described. This method represents an attempt to compare diverse interconnection network designs in a way that allows not only the best of existing designs to be identified but to suggest other, perhaps hybrid, networks that may offer better performance. Stepwise linear regression is used to develop a polynomial surface representation of performance in a (k+1) space with a total of k quantitative and qualitative independent variables describing graph-theoretic characteristics such as size, average degree, diameter, radius, girth, node-connectivity, edge-connectivity, minimum dominating set size, and maximum number of prime node and edge cutsets. Dependent variables used to measure performance are average message delay and the ratio of message completion rate to network connection cost. Response Surface Methodology (RSM) optimizes a response variable from a polynomial function of several independent variables. Steepest ascent path may also be used to approach optimum points
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