EXPOSED LAYOUTS OF THE BUTTERFLY NETWORK

2001 ◽  
Vol 02 (02) ◽  
pp. 233-247 ◽  
Author(s):  
AMI LITMAN
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Input Output ◽  
Layout Problem ◽  
Input And Output ◽  
Main Technique ◽  
Butterfly Network ◽  
N Input ◽  
Open Question ◽  
Minimal Area

This work studies layouts of the Butterfly network under the restriction that its input and output verticles are exposed, that is, are placed on the boundary of the grid. Avior et al. have shown that the minimal area of a layout of the n-input Butterfly, without the above restriction, is (1 + o(1))n2). The effect of this restriction on the area was left as an open question. This paper reveals that exposing the input and output vertices is essentially free. That is, it presents an exposed layout of the Butterfly having the same area as above. In this layout, the fractions of the input and output vertices assigned to each side of the grid are as follows when scanning the sides in a circular fashion: half of the inputs, half of the outputs, half of the inputs, and half of the outputs. We refer to such a layout as a [Formula: see text]-layout. The main technique employed in this layout is the reduction of the layout problem of the Butterfly to a certain layout problem of a complete bipartite graph. We use the same technique to produce a (I, 0, O, 0)-layout (inputs on one side and outputs on the opposite side) in area 2 + o(1))n2. Finally, we show that the area of a [Formula: see text]-layout is greater than [Formula: see text]. Hence, this input-output configuration is more area-demanding than [Formula: see text].

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Recessive Transition Mechanism of Arable Land Use Based on the Perspective of Coupling Coordination of Input–Output: A Case Study of 31 Provinces in China

Land ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 41
Author(s):  
Yi Lou ◽  
Guanyi Yin ◽  
Yue Xin ◽  
Shuai Xie ◽  
Guanghao Li ◽  
...  
Keyword(s):  
Land Use ◽  
Evaluation Model ◽  
Arable Land ◽  
Temporal Trends ◽  
Mainland China ◽  
Input Output ◽  
Transition Mechanism ◽  
Input And Output ◽  
Spatio Temporal ◽  
Arable Land Use

In the rapid process of urbanization in China, arable land resources are faced with dual challenges in terms of quantity and quality. Starting with the change in the coupling coordination relationship between the input and output on arable land, this study applies an evaluation model of the degree of coupling coordination between the input and output (D_CCIO) on arable land and deeply analyzes the recessive transition mechanism and internal differences in arable land use modes in 31 provinces on mainland China. The results show that the total amount and the amount per unit area of the input and output on arable land in China have presented different spatio-temporal trends, along with the mismatched movement of the spatial barycenter. Although the D_CCIO on arable land increases slowly as a whole, 31 provinces show different recessive transition mechanisms of arable land use, which is hidden in the internal changes in the input–output structure. The results of this study highlight the different recessive transition patterns of arable land use in different provinces of China, which points to the outlook for higher technical input, optimized planting structure, and the coordination of human-land relationships.

Voice Input/Output in Perspective

1982 ◽  
Vol 26 (3) ◽  
pp. 218-222
Author(s):  
Ken Funk ◽  
Ed McDowell
Keyword(s):  
Input Output ◽  
Voice Input ◽  
Input And Output ◽  
Machine Communication

Lately, quite a lot of effort has been put into the development of voice input and output systems for human-machine communication. In this paper, we point out that while voice I/O is ideal for many applications, there are others for which it is ill-suited and there is a danger that fascination with the technology may well result in its misuse.

scholarly journals An Exact Turán Result for Tripartite 3-Graphs

10.37236/5203 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Adam Sanitt ◽  
John Talbot
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Maximum Size ◽  
Unique Graph

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. We prove an analogue of this result for 3-graphs. Let $K_4^-=\{123,124,134\}$, $F_6=\{123,124,345,156\}$ and $\mathcal{F}=\{K_4^-,F_6\}$: for $n\neq 5$ the unique $\mathcal{F}$-free 3-graph of order $n$ and maximum size is the balanced complete tripartite 3-graph $S_3(n)$ (for $n=5$ it is $C_5^{(3)}=\{123,234,345,145,125\}$). This extends an old result of Bollobás that $S_3(n) $ is the unique 3-graph of maximum size with no copy of $K_4^-=\{123,124,134\}$ or $F_5=\{123,124,345\}$.

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