# Combinatorics

9
How can I optimize a function on a Planar Graph?

Assuming we have a planar Graph G=(V,E). There is a function f(G) which returns a positive real number and we wish to assign either 0 or 1 to each node in order to minimize the function f(G). Trying all different combinations of assigning 0 or 1 to each node to find the combination which returns the minimum value of f(G) would be computational intractable. Hence, the question is whether there exists a algorithm to find (or approximate) the minimum for such a function.

Note that the function f has the properties that a solution exists and is unique. Further, it is known that the Graph that minimizes the function has 0 in the vast majority of nodes and the set where the nodes are 1 is connected.

A couple of other quick thoughts.

1.  You could view your function as a set function, which depends only on the set of vertices selected.  If you could prove that the function has some nice property, such as submodularity, then you might be able to make progress without using the graph at all.

2. If you could show that your function can be well-approximated with a polynomial, then you could convert your problem into a polynomial optimisation problem.  Then you could use a software package for polynomial optimisation, such as GloptiPoly. (Note that x_i is binary if and only if x_i = x_i^2.)

2
How to apply combinatorial analysis in image fusion?

Is there any provision to apply combinatorial analysis on image fusion?

• Source
##### Article: Medical Image Fusion in Wavelet and Ridgelet Domains: A Comparative Evaluation
[Hide abstract]
ABSTRACT: Medical image fusion facilitates the retrieval of complementary information from medical images and has been employed diversely for computer-aided diagnosis of life threatening diseases. Fusion has been performed using various approaches such as Pyramidal, Multi-resolution, multi-scale etc. Each and every approach of fusion depicts only a particular feature (i.e. the information content or the structural properties of an image). Therefore, this paper presents a comparative analysis and evaluation of multi-modal medical image fusion methodologies employing wavelet as a multi-resolution approach and ridgelet as a multi-scale approach. The current work tends to highlight upon the utility of these approaches according to the requirement of features in the fused image. Principal Component Analysis (PCA) based fusion algorithm has been employed in both ridgelet and wavelet domains for purpose of minimisation of redundancies. Simulations have been performed for different sets of MR and CT-scan images taken from ‘The Whole Brain Atlas'. The performance evaluation has been carried out using different parameters of image quality evaluation like: Entropy (E), Fusion Factor (FF), Structural Similarity Index (SSIM) and Edge Strength (QFAB). The outcome of this analysis highlights the trade-off between the retrieval of information content and the morphological details in finally fused image in wavelet and ridgelet domains.
International Journal of Rough Sets and Data Analysis 07/2015; 2(2):78-91. DOI:10.4018/IJRSDA.2015070105
• Antonio Piccolomini d'Aragona added an answer:
4
Can someone help on an exercise on the consistency of the SKI calculus?

The exercise I'm dealing with asks me to show that by adding S = K to the usual reduction rules for the SKI-calculus, one obtains an inconsistent equivalence. This must be done without using Böhm's theorem.

Now, I've found two terms (not combinators) M and N with the following properties:

M = x
N = y
M can be obtained from N by replacing one or more occurrences of S with K

From the rule S = K we thereby get that M = N, and hence that x = y. Which means that any term can be proved equal to any other.

Do you think such an answer could work? Actually, Böhm's theorem (as shown in the book I am studying) establishes that, for distinct combinators G and H, there is a combinator D such that DxyG = x and DxyH=y. So, I feel it would have been more appropriate to find a combinator D such that Dx1x2S = x_i and Dx1x2K = x_j with i , j = 1 , 2 and i ≠ j. But I have HUGE problems in finding combinators, so I have only found terms respectively reducing to one of the variables, and obtainable one from another by replacing S with K or vice versa.

Yeah I see now there's a much simpler solution. But I wondered if I had "well done" the exercise. Actually, it seems to me it works. I found two terms, M and N, the first one reducing to an arbitrary x and the second one to an arbitrary y. Moreover, M can be obtained from N by replacing an occurrence of S with K and therefore, as soon as one admits S = K as a rule, it follows that M = N. Since M = x and N = y, by transitivity one should get x = y for every x and y.

Actually, M and N, as I built them, contain other terms which are in the end not relevant to respective reducibility to x and y. But that's should not be a problem. It should be like to have two terms with variables (x, y, z, w), possibly with z = w, respectively reducing to x and y.

That's how I did the exercise...

PS: inconsistent in the sense that one can trivialize the equality relation.

• Marcel Van de Vel added an answer:
16
Does the discrete n-circle (n even) admit a partition into n/2 pairs, all with a distinct diameter?

A (discrete) n-circle is the set of complex n-th roots of unity, or: the vertices of a regular n-gon. The above question arose as part of a (nearly finished) research project on a method to produce unpredictable number sequences. Although my partial answers are no longer needed for the project, the simple-looking and still unsettled problem keeps intriguing me.

I proved that if a partition exists into pairs of distinct diameters, then n must be of type 8k or 8k+2 (k>0 integer). Computer generated examples confirm that for n <= 112, these types are *exactly* the sizes that work. The computer was stopped after running for two days on the case n=114 (having inspected nearly 0.000...001% (about 300 zeros) of the total search space). The only hope on further information must come from construction methods other than brute-force search with back-tracking and from proofs.

Specifically, the problem becomes this: Design an algorithm that is guaranteed to produce a partition (as desired) whenever there exists one and reports failure otherwise. Unlike the current backtracking brute-force search, the algorithm should provide answers in a reasonable time [Added 09-12-2013: solved]. The problem is certainly NP (Nondeterministic Polynomial), but chances are that it is NP-complete [ Added 09-12-2013: not NP complete].

A weaker problem is to find the largest number b <= n/2 such that *any* b vertex pairs with different diameters can be rotated apart in the n-circle for *any* (even) n. It might be "(n/2)-1", I haven't checked on this yet. Ultimately, one should be able to determine the best b for each individual n (including the odd case) [ Added 09-12-2013: this is still wide open. Exhaustive computer search is getting quite demanding, even for fairly low n ].

[ State of affairs 30-03-2015 ]  The maximal number of pairs (not necessarily with distinct diameters) that can be rotated apart in an n-gon has been determined (with some computer assistance) for n = 8--12, 14, 15, 18, 20, 21, 26.  Another problem that kept me busy lately is this: given n = t*m^2 with t, m >= 2, is it true that t+1 sets of size m can be rotated apart in the n-circle? For n=32 (case t=2, m=4) I already found a (rather elaborate) solution in 2012. Recently I found a promising new approach involving cyclotomic polynomials, providing a shorter proof for n=32 and , in fact, an elegant proof for all cases of type n = 2*m^2 (until recently, m needed to be prime).

A revised version of my paper is available on request. It contains another solved problem on my wish list: given n = t m2 with t >= 2 and m >=2, every t+1 sets of size m can be rotated apart in a regular n-gon. One application of this relates with the sharpness of an older result which is described in a pdf extract of my paper (see the attached file).

if we take t=2 and m=8 (so n=128), we see that any three sets of size 8 can be rotated apart. The problem I need to solve is this:

find three sets of size 8 in a regular 127-gon which cannot be rotated apart.

A positive answer to this proves two distinct results to be sharp.

• David G Glynn added an answer:
2
Can we have any logical definition of infinite matroids and related defined codes?

recently, some researchers work on infinite structure of matroids.

independent axiom of matroids  have a first role in definition of codes on GF(q) that we can see this point in representable matroids. now , can we have any logical definition  of this point for related codes of infinite matroids?

In general, there would be two kinds of infinity for matroids: either infinite number of points or infinite rank (dimension), or maybe both.  For finite matroids, there are various axiom systems involving e.g. rank function (of subsets of points), independent sets, bases, spanning sets, circuits etc, which people like Brylawski called "cryptomorphic" axiom sets, that are all equivalent to each-other.  Infinite matroids could easily be constructed e.g. by repeating points in PG(n,q) an infinite number of times.  To obtain infinite dimensional matroids (that are representable) one would need an infinite dimensional vector space over a field and then an infinite spanning set of point with the induced rank function or independent sets and so on.  To get codes of length n and dimension k one normally takes a set S of n points in PG(k-1,q) (or vector space of rank k) so that the matroid structure will correspond to the code.  (Hyperplanes h of the matroid correspond to codewords.  The places on the word that are non-zero correspond to points of S\h.  The zeros are the points in the intersection of S and h.)  This would be the same if one wanted infinite dimensions.  Take an infinite dimensional vector space over GF(q) (or any field) and then an infinite number of points would induce a code structure.  There are actually three possible infinities there: dimension, number of points, size of field.

4
Do you know the number of all spanning trees of a given class?

When a bipartite complete graph Km,n is given, two subgraphs of Km,n are in the same class when the degree of each right vertex coincides. I want to know the number of all spanning trees in a given class.

Any spanning tree in Km,n has M+N-1 edges. A class whose right degrees do not sum up to M+N-1 does not contain any spanning tree. The number of classes with total degree M+N-1  is the repeated combination of country labels taken N -1 times. Thus the number take the form

(M + N - 2)! / (M - 1)! (N -1)! .

From Scoins' formula the number of all spanning trees in Km,n is

MN-1NM-1 .

As a consequence, there are in general many spanning trees in a class in which the right degrees sum up to M+N-1.  I want to know an explicit formula that gives the number of all spanning trees for a given class with degree sum M + N -1.

This question is derived in the course of Ricardian trade theory study.

Dear Robert A. Beeler,

thank you for your comment. I am expecting to hear from you good news.

Shiozawa

12
Can we use discrete mathematics in modeling of control and dynamical systems?

We usually use differential equations, ordinary and partial, difference and delayed. But, could dynamics be captured using discrete mathematics structures, or combinatorics?

Thanks. (:

9
Is there a bijective proof of : the rational number q_{m,n} = \frac {(2m)! (2n)!} {m! n! (m+n)!}, where m,n are positive integers, is an integer ?

I found this problem without comments in a french exercise book of 1982, now out of print : {\em 1932 exercices de mathématiques} by Luc Moisotte, ISBN 2-04-015483-3.

Thank you, Anna,

but I look for a possible "bijective" proof, I mean :
this number is an integer because it counts something than we can describe.

Sincerly, Jean

2
Re: research
dear friends. Presently I am doing research in automata theory. What is the latest research going on automata theory. friends, I am now in confusion. how to start and how it grow because I am basically mathematician. I want to know more ideas. also Is there any relavant materials available net. actually I started with buchi automaton in automata theory. but I do not know proceed that I have seen many materials. please help me & suggest me good ideas.
thanks & regards
O.V.SHANMUGA SUNDARAM
ovs3662@gmail.com

we completed ph.d thesis work. now preparing synopsis. for this we need external foreign examiner for evaluation of my ph.d thesis. probably we submit thesis before December 2015.. Our topic is related to automata theory using graph theory concept in the core chapter. we request you kindly know or interested please send your latest profile... thank you for considering my obligation....to my email-id is ovs3662@gmail.com

• SIMON RAJ F added an answer:
14
Is there any Barnette Graph with 2k (even number greater than 84) vertices?

In 1969 , David Barnette conjectured that 3 regular , 3-connected , bipartite , planar graph is Hamiltonian. . I am interested  to generate Barnette  graph for given  even number of vertices. There are countably finite number of Barnette graphs available in the literature.

Consider a grid graph with 4 vertices (cycle C 4) which is Hamiltonian. Increasing dimension in one direction , we see that the resulting graph is always Hamiltonian but not Barnette. Can one generate  countably infinite number of  Barnette graphs from one small Barnette graph?

Increasing dimension of the graph or network. like Hexagonal network.

7
What are the applications of additive combinatorics in the field of engineering and architecture?

I would like to know some practical applications of additive combinatorics in the field of engineering and architecture. Please suggest some useful reference.

To enable agility by additive manufacturing (complexity, agility, efficiency) implementing concurrent, hybrid processes, considering the design space with topology organization,
production constraints, and optimization of variables and materials. Innovation is better through additive manufacturing, such as using Additive Topology Optimized Manufacturing (ATOM) which helped place brackets on the Black Hawk helicopter, and drastically reduces wasted material.

3
Baker Hausdorff decoupling formula
I have a two mode squeezed state ,
exp(ab-a^+b^+)|0>-a|0>_b . I would like to show entangled form which it will give. In this case [A,[A,B]] !=0 & [B,[A,B]] !=0 (!= means not equal) .therefore I can not use well known Glauber (Baker Hausdorff) decoupling formula directly. Does someone has experience on this problem or good suggestion for me. I think, I can do Taylor expansion then try to find some way to decouple this, I hope it will work ...

I think I misunderstood your question.

• Miguel A. Pizaña added an answer:
4
What are the major real life or practical applications of intersection graphs?

What are the major real life or practical applications of intersection graphs? Please suggest some good reference materials.

Clique Graphs, a special case of intersection graphs, have been used in Loop Quantum Gravity:

M. Requardt.
(Quantum) spacetime as a statistical geometry of lumps in random networks.
Classical Quantum Gravity 17 (2000) 2029--2057.

M. Requardt.
Space-time as an order-parameter manifold in random networks and the emergence of physical points.
In Quantum theory and symmetries (Goslar, 1999), pages
555--561. World Sci. Publ., River Edge, NJ, 2000.

M. Requardt.
A geometric renormalization group in discrete quantum space-time.
J. Math. Phys. {\bf 44} (2003) 5588--5615.

• Sam W. Murphy added an answer:
8
Knapsack Packing Problem. Anyone have experience with this?

I am trying to calculate the most compact way of grouping a set of pixels together. Does anyone have a readable guide on how to do this?

My initial results are given below for clusters of up to 10 pixels. Results are expressed in terms of the sum of unique interpixel distances for a given cluster (e.g. for a 3 pixel cluster it is the sum of the the distances ab, ac, bc).

1 = 0

2 = 1

3 = 3.4

4 = 6.8

5 = 13.5

6 = 21.1

7 = 31.4

8 = 44.1

9 = 58.9

10 = 78.5

Thank you Artiom you have been very helpful! (I have only just seen your previous post now, I must have posted mine at the same time!).. I will keep working on this problem

6
Just where do we draw the distinctive line between traditional data analysis and present day (Big) data analytics?

Some authorities like Davenport have already explained that traditional (Small?) data relates to corporate operations while Big data relates to corporate products and services.

The size of what people mean when they say big data changes (and varies by discipline), so 20 years ago it would have been a lot smaller than now. It might be what you can't store (or do simple analysis of) on a new desktop. Here is a quote from a White House report:

There are many definitions of “big data” which may differ depend ing on whether you are a computer scientist, a financial analyst, or an entrepreneur pitch
ing an idea to a venture capitalist. Most definitions reflect the growing technological ability to capture, aggregate, and process an ever-greater volume, velocity, and variety of data.

https://www.whitehouse.gov/sites/default/files/docs/big_data_privacy_report_may_1_2014.pdf

So, imo, there is no line (and if someone drew one today, its wrong by tomorrow).

10
Where can I find useful literature on graph theoretical applications to biological networks?

Please provide me information regarding the recent developments in the mathematical, especially graph theoretical, studies on biological networks. Please give some good reference too..

Thanking you in advance,

Sudev

Thank you so much Professor Ljubomir Jacić, I will go through that.

• Marzieh Eini Keleshteri added an answer:
2
What can be said about the $q$-analogue of the Sheffer identity?

In \cite{Roman} page 25 we read that,  a sequence $s_n(x)$ is Sheffer for $(g(t), f(t))$, for some invertible $g(t)$, if and only if

$$s_n(x+y)=\sum\limits_{k=0}^{\infty}\binom nk p_k(y) s_{n-k}(x),$$

for all $y$ in complex numbers,  where $p_n(x)$ is associated to $f(t)$.

Noting to the fact that $e_q(x+y) \neq e_q(x)e_q(y)$, leads to conclude that $s_{n,q}(x+y) \neq e_q(yt)s_{n,q}(x)$, and ,therefore, we do not have the $q$-analogue of the identity above directly. Is it possible to express the $q$-analogue of the above mentioned identity in any other way, or should we neglect such an identity for $q$-Sheffer sequences at all?

Any contribution is appreciated in advance.

\bibitem{Roman}Roman S., Rota G. The umbral calculus. Advances Math. 1978;27:95–188.

Dear Dr. Waldemar W Koczkodaj

Thanks for your response. I faced with this question, for the first time, while I was studying the sequence of $q$-Appell polynomials. Since that time, this question has been in my mind for a long time and so far I have not been able to make myself convinced by a perfect answer to it. The reason to ask this question here, actually, is to consult with the experts of this field and read about their different ideas which are originated, clearly, from different points of view. Although I am enthusiastic for learning more and more and go forward through the science, in case that I found a good answer which helps me to publish my studies, I make you sure that I will definitely obey the publication rules and humanity.

9
What is the number of subsets of a finite set of non-negative integers which are neither the sumsets or the summands of other subsets of $X$?

Suppose that X is a finite set of positive integers. The sumset of two subsets A and B of X is defined as A+B={a+b:a\in A, b\in B}. Then, what is the number of subsets of X which are neither the non-trivial sumsets of any two other subsets of X nor the non-trivial summands of any other subsets of X?  Also, please suggest some useful references in this area.

Thanking you in advance,

Sudev

I'm still trying to understand the question.

Suppose X = {1,2,3,5}.

Then X has 2^4 possible subsets.  But {2,3} is out because it is the sumset of {1,2} and {1}.  Then are {1,2} and {1] out because they are summands of {2,3}?

If so, we're left with just two subsets - {1,3,5} and {1, 2, 3, 5}.

9
What are the graphs whose total graphs are complete graphs?

The total graph T(G) of a given graph G is a graph such that the vertex set of T corresponds to the vertices and edges of G and two vertices are adjacent in T if their corresponding elements are either adjacent or incident in G. Can we have non-trivial graphs whose total graphs are complete?

I checked  response of Prof. Singaram Dharmalingam now, quick proof :)

Thank you for the question though.

Good Luck!

• Qefsere Doko Gjonbalaj added an answer:
4
How can we relate set theory with networks theory?

Would you please help me to identify some applications of set theory, in particular sumset theory, in networks such as  in social and biological networks? Please suggest some useful reading too/

I will suggest you this link

Connected Dominating Set: Theory and Applications - Springer
http://www.springer.com/mathematics/applications/book/978-1-4614-5241-6

6
Can anybody suggest any references for combinatorial studies on perturbation theory?

Are there any combinatorial studies on perturbation theory? Can we relate graph theory to that area? If so, please suggest some useful references for a beginner like me.

Thank you Prof. Sudev.

8
Is it right to take the sumset A + ∅ = ∅?

Is the concept A + ∅ = ∅ correct? A is any set of non-negative integers. I
think that if it is so, it contradicts the condition on the cardinality of sum sets
that |A| + |B| − 1 ≤ |A + B| ≤ |A| |B|. Please give your expert opinions.

I guess, Kemperman's theorem is for non-empty sets.

18
Is there any possibility for the sumset of two sets of integers, which are not arithmetic progressions, to be an arithmetic progression?

The notion of sumset of two sets is defined as A+B={a+b:a in A, b in B}. If the elements of A and B are not in AP,  then can the elements of A+B be an arithmetic progression? If so, what are the conditions required for that? Please can you recommend me some good references.

Thanking you in advance,

Don't know if that was already pointed out, but the example by Alex Ravsky is as "bad" as that neither A nor B contain a an arithmetic progression of length 3.

4
Why does the four colorability of planar graphs not ensure the non-biplanarity of K_9?

My earlier expectation regarding the sufficiency of 4CT to adress this issue, is not correct.

• Source
##### Article: An Analytic Proof of Four Color Problem
[Hide abstract]
ABSTRACT: Abstract – An analytical proof of the Four Color Conjecture has been described in this article. Kempe’s chain argument and Heawood’s technique to prove Five color theorem' has been exploited. Success has come through the searching of special triangles, around a vertex of degree 5, for three recursions. This proof stands on the principle of mathematical induction, so requires no computer assistance.

+ 1 more attachment

My latest publication in http://www.ijeijournal.com/pages/v4i6.html may help to give some insight on this issue.

4
Compute the number of path from a node to a different one in a weighted undirected connected 3-regular graph.
The graph is a weighted undirected connected 3-regular graph. The number of nodes is N.
For each node there is one loop with weight $= \frac{1}{N}, and two other edges which goes from the node to its two `nearest neighburs with weight$= \frac{\epsilon}{N}, where \^epsilon is a small parameter.
Therefore, we would like to know for a given value of $N$ how many different paths of lenght $2T$ are possible to go from one node to another passing through $k$ edges with weight $= \frac{\epsilon}{N}$ ?

I agree with Prof. Patrick Solé.  Maple is a very useful software for any kind of mathematical operations.

The number of paths of length n on the pair of given vertices (vi, vj) is the (i, j)th entry of An where A is the adjacency matrix of the given graph G.

You may refer refer the following link.

B Roberts, D P Kroese, Estimating the Number of s-t Paths in a Graph, http://jgaa.info/accepted/2007/RobertsKroese2007.11.1.pdf

Also visit the following page for a recursive algorithm to find all paths between two given vertices of a given graph.  http://www.technical-recipes.com/2011/a-recursive-algorithm-to-find-all-paths-between-two-given-nodes/

3
Are there any specific studies on the maximal bipartite sub-graphs of different products of two regular graphs?

Could you suggest me some of the articles on the maximal bipartite sub-graphs of different products of graphs, especially two regular graphs?

Dear Prof. Vitaly Voloshin, Thank you for the links...

4
Where can I find some real life or scientific applications of the sumsets of two sets of real numbers?

Please provide me some practical/real life applications for the sumsets of the sets of integers or real numbers. Please suggest me some useful references too..

The sum is sometimes a direct sum, like in coordinate axes. R2 = (1,0) R + (0,1) R and this fact has a lot of applications (!) Another interesting fact is that R = Z + [0,1] and the decomposition is again unique. This leads to the functions integer part and fractional part. Also interesting, the sum Z + sqrt(2) Z is dense in R. These are the most easy (trivial) examples, but there are a lot of nice sumsets of reals!

1
Is it possible to find an infinite arithmetic progression of value of k such that we can get primes of the form 6k+1 successively?

for k=1, 2, 3 we get 7, 13, 19 as primes.

for k=5,  6, 7 we get  31, 37,  43,  as primes.

Even if true this is likely beyond current technology. It would imply Green Tao theorem:

http://en.wikipedia.org/wiki/Primes_in_arithmetic_progression

3
How to decrease the bounded gaps between primes?

Last year, Dr. Yitang Zhang has published a paper for the upper bound of twin primes, which is 7*10^7. Does anyone has idea to achieve lower bounded gap?

Actually, it is known, conditionally on GRH if I remember correctly, that at least one of the following hold holds true:
i) p_{n+1}-p_n =2  infinitely often
ii) p_{n+1}-p_n =4 infinitely often
iii) p_{n+1}-p_n =6 infinitely often

[See this beautiful lecture https://www.youtube.com/watch?v=pp06oGD4m00 ]