scholarly journals An Efficient Algorithm to Count Tree-Like Graphs with a Given Number of Vertices and Self-Loops

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 923 ◽  
Author(s):  
Naveed Ahmed Azam ◽  
Aleksandar Shurbevski ◽  
Hiroshi Nagamochi
Keyword(s):  
Dynamic Programming ◽  
Graph Theory ◽  
Lower Bound ◽  
Upper Bound ◽  
Efficient Algorithm ◽  
Rooted Tree ◽  
Chemical Compounds ◽  
Natural Sciences ◽  
Interesting Problem ◽  
Cycle Rank

Graph enumeration with given constraints is an interesting problem considered to be one of the fundamental problems in graph theory, with many applications in natural sciences and engineering such as bio-informatics and computational chemistry. For any two integers n≥1 and Δ≥0, we propose a method to count all non-isomorphic trees with n vertices, Δ self-loops, and no multi-edges based on dynamic programming. To achieve this goal, we count the number of non-isomorphic rooted trees with n vertices, Δ self-loops and no multi-edges, in O(n2(n+Δ(n+Δ·min{n,Δ}))) time and O(n2(Δ2+1)) space, since every tree can be uniquely viewed as a rooted tree by either regarding its unicentroid as the root, or in the case of bicentroid, by introducing a virtual vertex on the bicentroid and assuming the virtual vertex to be the root. By this result, we get a lower bound and an upper bound on the number of tree-like polymer topologies of chemical compounds with any “cycle rank”.

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

scholarly journals Learning Immune-Defectives Graph through Group Tests

2015 ◽  
Author(s):  
Abhinav Ganesan ◽  
Sidharth Jaggi ◽  
Venkatesh Saligrama
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
High Probability ◽  
Chemical Compounds ◽  
Pathogen Identification ◽  
Pooling Design ◽  
Two Stage ◽  
Decoding Algorithms ◽  
Lead Compounds ◽  
Multiplicative Factor

This paper deals with an abstraction of a unified problem of drug discovery and pathogen identification. Here, the ``lead compounds'' are abstracted as inhibitors, pathogenic proteins as defectives, and the mixture of ``ineffective'' chemical compounds and non-pathogenic proteins as normal items. A defective could be immune to the presence of an inhibitor in a test. So, a test containing a defective is positive iff it does not contain its ``associated'' inhibitor. The goal of this paper is to identify the defectives, inhibitors, and their ``associations'' with high probability, or in other words, learn the Immune Defectives Graph (IDG). We propose a probabilistic non-adaptive pooling design, a probabilistic two-stage adaptive pooling design and decoding algorithms for learning the IDG. For the two-stage adaptive-pooling design, we show that the sample complexity of the number of tests required to guarantee recovery of the inhibitors, defectives and their associations with high probability, i.e., the upper bound, exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of items. For the non-adaptive pooling design, in the large inhibitor regime, we show that the upper bound exceeds the proposed lower bound by a logarithmic multiplicative factor in the number of inhibitors.

scholarly journals Improved bounds on the crossing number of butterfly network

2013 ◽  
Vol Vol. 15 no. 2 (Graph Theory) ◽  
Author(s):  
Paul D. Manuel ◽  
Bharati Rajan ◽  
Indra Rajasingh ◽  
P. Vasanthi Beulah
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Upper Bound ◽  
Crossing Number ◽  
Previous Estimate ◽  
Butterfly Network ◽  
International Audience

Graph Theory International audience We draw the r-dimensional butterfly network with 1 / 44r+O(r2r) crossings which improves the previous estimate given by Cimikowski (1996). We also give a lower bound which matches the upper bound obtained in this paper.

scholarly journals Lower Bounds for the Cop Number when the Robber is Fast

2011 ◽  
Vol 20 (4) ◽  
pp. 617-621 ◽  
Author(s):  
ABBAS MEHRABIAN
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Lower Bounds ◽  
Regular Graph ◽  
Upper Bound ◽  
Upper Bounds ◽  
Cops And Robbers ◽  
Vertex Graph

We consider a variant of the Cops and Robbers game where the robber can movetedges at a time, and show that in this variant, the cop number of ad-regular graph with girth larger than 2t+2 is Ω(dt). By the known upper bounds on the order of cages, this implies that the cop number of a connectedn-vertex graph can be as large as Ω(n2/3) ift≥ 2, and Ω(n4/5) ift≥ 4. This improves the Ω($n^{\frac{t-3}{t-2}}$) lower bound of Frieze, Krivelevich and Loh (Variations on cops and robbers,J. Graph Theory, to appear) when 2 ≤t≤ 6. We also conjecture a general upper boundO(nt/t+1) for the cop number in this variant, generalizing Meyniel's conjecture.

scholarly journals On Winning Fast in Avoider-Enforcer Games

10.37236/328 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
János Barát ◽  
Miloš Stojaković
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Complete Graph ◽  
Upper Bound ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Additive Constant ◽  
Main Interest ◽  
Free Graph ◽  
Positional Games

We analyze the duration of the unbiased Avoider-Enforcer game for three basic positional games. All the games are played on the edges of the complete graph on $n$ vertices, and Avoider's goal is to keep his graph outerplanar, diamond-free and $k$-degenerate, respectively. It is clear that all three games are Enforcer's wins, and our main interest lies in determining the largest number of moves Avoider can play before losing. Extremal graph theory offers a general upper bound for the number of Avoider's moves. As it turns out, for all three games we manage to obtain a lower bound that is just an additive constant away from that upper bound. In particular, we exhibit a strategy for Avoider to keep his graph outerplanar for at least $2n-8$ moves, being just $6$ short of the maximum possible. A diamond-free graph can have at most $d(n)=\lceil\frac{3n-4}{2}\rceil$ edges, and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$ is small compared to $n$, we show that Avoider can keep his graph $k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number of edges a $k$-degenerate graph can have.

scholarly journals The Leaf Function of Graphs Associated with Penrose Tilings

2020 ◽  
Vol 1 (1) ◽  
pp. 1-24
Author(s):  
Carole Porrier
Keyword(s):  
Graph Theory ◽  
Lower Bound ◽  
Upper Bound ◽  
Dimensional Space ◽  
Geometric Properties ◽  
3 Dimensional ◽  
Numerical Errors ◽  
Computation Efficiency ◽  
Number Of Leaves ◽  
Penrose Tilings

In graph theory, the question of fully leafed induced subtrees has recently been investigated by Blondin Massé et al in regular tilings of the Euclidian plane and 3-dimensional space. The function LG that gives the maximum number of leaves of an induced subtree of a graph $G$ of order $n$, for any $n\in \N$, is called leaf function. This article is a first attempt at studying this problem in non-regular tilings, more specifically Penrose tilings. We rely not only on geometric properties of Penrose tilings, that allow us to find an upper bound for the leaf function in these tilings, but also on their links to the Fibonacci word, which give us a lower bound. Our approach rely on a purely discrete representation of points in the tilings, thus preventing numerical errors and improving computation efficiency. Finally, we present a procedure to dynamically generate induced subtrees without having to generate the whole patch surrounding them.

scholarly journals On Ensuring Multilayer Wirability by Stretching Layouts

VLSI Design ◽  
1998 ◽  
Vol 7 (4) ◽  
pp. 365-383
Author(s):  
Teofilo F. Gonzalez ◽  
Si-Qing Zheng
Keyword(s):  
Dynamic Programming ◽  
Lower Bound ◽  
Upper Bound ◽  
Area Increase ◽  
Different Types ◽  
Np Complete ◽  
Layout Area ◽  
The One

Every knock-knee layout is four-layer wirable. However, there are knock-knee layouts that cannot be wired in less than four layers. While it is easy to determine whether a knock-knee layout is one-layer wirable or two-layer wirable, the problem of determining three-layer wirability of knock-knee layouts is NP-complete. A knock-knee layout may be stretched vertically (horizontally) by introducing empty rows (columns) so that it can be wired in fewer than four layers. In this paper we discuss two different types of stretching schemes. It is known that under these two stretching schemes, any knock-knee layout is three-layer wirable by stretching it up to (4/3) of the knock-knee layout area (upper bound). We show that there are knock-knee layouts that when stretched and wired in three layers under scheme I (II) require at least 1.2 (1.07563) of the original layout area. Our lower bound for the area increase factor can be used to guide the search for effective stretching-based dynamic programming three-layer wiring algorithms similar to the one presented in [8].

An efficient route design for solid waste collection using graph theory and the algorithm of the traveling agent in dynamic programming

2021 ◽  
Vol 193 (5) ◽  
Author(s):  
Moisés Filiberto Mora Murillo ◽  
Walter Alfredo Mora Murillo ◽  
Luis Xavier Orbea Hinojosa ◽  
Arlys Michel Lastre Aleaga ◽  
Gabriel Estuardo Cevallos Uve ◽  
...  
Keyword(s):  
Dynamic Programming ◽  
Graph Theory ◽  
Solid Waste ◽  
Waste Collection ◽  
Solid Waste Collection ◽  
Route Design ◽  
Efficient Route

scholarly journals Multiple closed orbits for N-body-type problems

1998 ◽  
Vol 58 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Shiqing Zhang
Keyword(s):  
Lower Bound ◽  
Periodic Solutions ◽  
Upper Bound ◽  
Critical Value ◽  
Body Type ◽  
Fixed Energy ◽  
Closed Orbits

Using the equivariant Ljusternik-Schnirelmann theory and the estimate of the upper bound of the critical value and lower bound for the collision solutions, we obtain some new results in the large concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planar N-body type problems.

Limit Cycle Bifurcations for Piecewise Smooth Hamiltonian Systems with a Generalized Eye-Figure Loop

2016 ◽  
Vol 26 (12) ◽  
pp. 1650204 ◽  
Author(s):  
Jihua Yang ◽  
Liqin Zhao
Keyword(s):  
Lower Bound ◽  
Limit Cycle ◽  
Hamiltonian Systems ◽  
Limit Cycles ◽  
Upper Bound ◽  
Melnikov Function ◽  
Piecewise Smooth ◽  
First Order ◽  
Period Annulus

This paper deals with the limit cycle bifurcations for piecewise smooth Hamiltonian systems. By using the first order Melnikov function of piecewise near-Hamiltonian systems given in [Liu & Han, 2010], we give a lower bound and an upper bound of the number of limit cycles that bifurcate from the period annulus between the center and the generalized eye-figure loop up to the first order of Melnikov function.

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