On Functions of K-Balanced Matroids

Talal Al-Hawary

Department of Mathematics, Yarmouk University, Irbid, Jordan.

DOI: 10.4236/ojdm.2017.73011                

Abstract

In this paper, we prove an analogous to a result of Erdös and Rényi and of Kelly and Oxley. We also show that there are several properties of k-balanced matroids for which there exists a threshold function.

Keywords

K-Balanced, Matroid, Projective Geometry, Threshold Function

1. Introduction

We begin with some background material, which follows the terminology and notation in [1] . Let $M=\left(E,F\right)$ denote the matroid on the ground set E with flats F. All matroids considered in this paper are loopless. In particular, if M is a matroid on a set E and X ⊆ E, then r(X) will denote the rank of X in M. We shall be considering projective geometries over a fixed finite field GF(q), recalling that

(see, for example [2] ) the number $\left[\begin{array}{c}r\\ n\end{array}\right]$ of rank-n subspaces of the projective

geometry PG (r − 1, q) is

$\frac{\left(q^r-1\right)\left(q^{r-1}-1\right)\cdots \left(q^{r-n+1}-1\right)}{\left(q^n-1\right)\left(q^{n-1}-1\right)\cdots \left(q-1\right)}.$

The uniform matroid of rank r and size n is denoted by $U_{r,n}$ where

$r=0,1,\cdots ,n$ . When r = n, the matroid $U_{r,r}$ is called free and when r = n = 0, the matroid $U_{0,0}$ is called the empty matroid. For more on matroid theory, the reader is referred to [1] - [15] . Let k be a nonnegative integer. The k-density of a

matroid M with rank greater than k is given by $d_k\left(M\right)=\frac{\left|M\right|}{r\left(M\right)-k}$ , where |M|

is the size of the ground set of M and r(M) is the rank of the matroid M. A matroid M is k-balanced if $r\left(M\right)>\left(k\left(k+1\right)\right)/2$ and

$d_k\left(M\right)\le d_k\left(M\right)$ (1)

for all non-empty submatroids $H⊑M$ and strictly k-balanced if the inequality is strict for all such H ≠ M. When k = 0, M is called balanced and when k = 1, M is called strongly balanced.

A random submatroid ${\omega }_r$ of the projective geometry $PG\left(r-1,q\right)$ is obtained from $PG\left(r-1,q\right)$ by deleting elements so that each element has, independently of all other elements, probability 1 − p of being deleted and probability 1 − p of being retained. In this paper, we take p to be a function p(r) of r. Let A be a fixed property which a matroid may or may not possess and $P_{r,p}\left(A\right)$ denotes the probability that ${\omega }_r$ has property A. We shall show that there are several properties A of k-balanced matroids for which there exists a function t(r) such that

$\underset{r\to \infty }{\lim }{P}_{r,p}\left(A\right)=\{\begin{array}{l}0,\underset{r\to \infty }{\lim }\frac{P}{t\left(r\right)}=0\\ 1,\underset{r\to \infty }{\lim }\frac{P}{t\left(r\right)}=\infty \end{array}$

If such a function exists, it is called a threshold function for the property A. For more on these notions, the reader is referred [16] [17] .

2. K-Balanced Matroids

In this section, we prove the following main result which is analogous to Theorem 1 of Erdös and Rényi [16] and to Theorem 1.1 of Kelly and Oxley [17] .

Theorem 1. Let m and n be fixed positive integers with n ≤ m and suppose that $B_{n,m}$ denote a non-empty set of k-balanced simple matroids, each of which have m elements and rank n and is representable over GF(q). Then a threshold function for the property B that ${\omega }_r$ has a submatroid isomorphic to some

member of $B_{n,m}$ is $q^{\frac{-rn}{m}}$ .

Proof. Let X and $B_{n,m}$ denote the number of submatroids of the matroid ${\omega }_r$ and $PG\left(n-1,q\right)$ respectively which are isomorphic to some member of $B_{n,m}$ . Then

$P_{r,p}\left(B\right)=P\left(X\ne 0\right)\le EX$

by definition of expectation. Therefore

$P_{r,p}\left(B\right)\le \left[\begin{array}{c}r\\ n\end{array}\right]B_{n,m}{p}^m\le B_{n,m}{p}^m{q}^{rn}\le B_{n,m}{\left(\frac{p}{q^{-\frac{rn}{m}}}\right)}^m$ .

Thus, if ${\lim }_{r\to \infty }\frac{p}{q^{-\frac{rn}{m}}}=0$ , then $\underset{r\to \infty }{\lim }{P}_{r,p}\left(B\right)=0$ .

Now suppose that ${\lim }_{n\to \infty }\frac{p}{q^{-\frac{rn}{m}}}=\infty$ . We need to show that, in this case, $\underset{n\to \infty }{\lim }{P}_{r,p}\left(B\right)=1$ . Let $D_{m,n}$ be the set of subsets A of $PG\left(r-1,q\right)$ for which the

restriction $PG\left(r-1,q\right)|A$ of $PG\left(r-1,q\right)$ to A is isomorphic to some member of $B_{n,m}$ . Then

$E{X}^2={\displaystyle {\sum }_{A_1\in D_{m,n}}{\displaystyle {\sum }_{A_2\in D_{m,n}}{p}^{\left|A_1\cup A_2\right|}}}={{\displaystyle \sum }}_{i=0}^m{p}^{m+i}{\propto }_i$ (2)

where ${\propto }_i$ equals the number of ordered pairs $\left(A_1,A_2\right)$ such that $A_1,A_2\in D_{m,n}$ and $\left|A_1\cap A_2\right|=m-i$ . Thus

$E{X}^2\le p^{2m}\left[{\left(B_{m,n}\left[\begin{array}{c}r\\ n\end{array}\right]\right)}^2+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}{\propto }_i\right].$

We now want to obtain upper bounds on the numbers ${\propto }_0,{\propto }_1,\cdots ,{\propto }_{m-1}$ , so suppose that $A_1,A_2\in D_{m,n}$ and $\left|A_1\cap A_2\right|=m-i$ where $0\le i\le m-1$ . Then as $PG\left(r-1,q\right)|A$ is k-balanced,

$\left(\left|A_1\cap A_2\right|\right)/\left(r\left(A_1\cap A_2\right)-k\right)\le m/\left(n-k\right)$

and so $r\left(A_1\cap A_2\right)\ge \left(\left(m-i\right)\left(n-k\right)\right)/m+k$ . It follows that

$\begin{array}{c}r\left(A_2\right)-r\left(A_1\cap A_2\right)\le n-\left(\left(m-i\right)\left(n-k\right)\right)/m-k\\ =\left(i\left(n-k\right)\right)/m\le \left(in\right)/m\end{array}$

and hence $r\left(A_2\right)-r\left(A_1\cap A_2\right)\le \lfloor \left(in\right)/m\rfloor$ where $\lfloor \left(in\right)/m\rfloor$ is the floor of $\left(in\right)/m$ .

Now ${\propto }_i={\beta }_i{\gamma }_i$ where ${\beta }_i$ is the number of ways to choose $A_1$ and ${\gamma }_i$ is the number of ways to choose $A_2$ so that $\left|A_1\cap A_2\right|=m-i$ , $A_1$ having already

been chosen. Clearly ${\beta }_i=B_{m,n}\left[\begin{array}{c}r\\ n\end{array}\right]$ . Once $A_1$ has been chosen, there are at

most $\left({}_{m-i}{}^m\right)$ choices for the subset $A_1\cap A_2$ of $A_1$ . Further, once $A_1\cap A_2$ has been chosen, $A_2$ must be contained in some rank n subspace W of PG(r-1,q) which contain the chosen set $A_1\cap A_2$ . The number δ of such subspaces W is bounded above by

$\left(\left(q^r-q^s\right)/\left(q-1\right)\right)\left(\left(q^r-q^{s+1}\right)/\left(q-1\right)\right)\cdots \left(\left(q^r-q^{n-1}\right)/\left(q-1\right)\right),$

where $s=r\left(A_1\cap A_2\right)$ . Thus $\delta \le q^{r\left(n-1\right)}$ . But it was shown above that

$n-s\le \lfloor \left(in\right)/m\rfloor$ ; hence $\delta \le q^{r\lfloor in/m\rfloor }.$ Once W has been chosen, there are at most $B_{m,n}$ choices for $A_2$ . We conclude that

${\gamma }_i\le \left({}_{m-i}{}^m\right)q^{r\lfloor in/m\rfloor }{B}_{m,n}$

and hence

${\alpha }_i\le \left[\begin{array}{c}r\\ n\end{array}\right]B_{m,n}^2\left({}_{m-i}{}^m\right)q^{r\lfloor in/m\rfloor }.$ (3)

Now as $EX=\left[\begin{array}{c}r\\ n\end{array}\right]B_{m,n}{p}^m,$ we have by Equation (2), that

$\frac{E{X}^2}{{\left(EX\right)}^2}\le 1+{\left(B_{m,n}\left[\begin{array}{c}r\\ n\end{array}\right]\right)}^{-2}+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}{\propto }_i$ .

Hence, by Equation (2),

$\frac{E{X}^2}{{\left(EX\right)}^2}\le 1+{\left(B_{m,n}\left[\begin{array}{c}r\\ n\end{array}\right]\right)}^{-2}+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}\left[\begin{array}{c}r\\ n\end{array}\right]B_{m,n}^2\left({}_{m-i}{}^m\right)q^{r\lfloor \frac{in}{m}\rfloor }.$ .

Thus $\frac{E{X}^2}{{\left(EX\right)}^2}\le 1+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}\left({}_{m-i}{}^m\right)\frac{q^{r\lfloor \frac{in}{m}\rfloor }}{\left[\begin{array}{c}r\\ n\end{array}\right]}\le 1+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}\left({}_{m-i}{}^m\right)\frac{q^{r\lfloor \frac{in}{m}\rfloor }}{q^{n\left(r-n\right)}}$

Since $\left[\begin{array}{c}r\\ n\end{array}\right]\ge q^{n\left(r-n\right)}$ . Thus

$\frac{E{X}^2}{{\left(EX\right)}^2}\le 1+{{\displaystyle \sum }}_{i=0}^{m-1}{p}^{i-m}{q}^{-rn+r\lfloor \frac{in}{m}\rfloor }\left({}_{m-i}{}^m\right)q^{n^2}.$ (4)

Now consider $p^{i-m}{q}^{-rn+r\lfloor \frac{in}{m}\rfloor }$ . We have

$q^{-rn+r\lfloor \frac{in}{m}\rfloor }\le q^{-r\left(n-\frac{in}{m}\right)}={\left(q^{rn/m}\right)}^{i-m}.$

Thus $p^{i-m}{q}^{-rn+r\lfloor \frac{in}{m}\rfloor }\le {\left(p{q}^{\frac{rn}{m}}\right)}^{i-m}$ . But ${\lim }_{r\to \infty }p{q}^{\frac{rn}{m}}=\infty ,$ hence ${\lim }_{r\to \infty }{\left(p{q}^{rn/m}\right)}^{i-m}=0$ for $0\le i\le m-1$ . It follows from Equation (4) that ${\lim }_{r\to \infty }\sup \frac{E{X}^2}{{\left(EX\right)}^2}\le 1$ ; hence ${\lim }_{r\to \infty }\frac{E{X}^2}{{\left(EX\right)}^2}=1$ . Therefore, by Chebyshev’s Inequality, ${\lim }_{r\to \infty }P\left(X\ne 0\right)=1$ . We conclude that $q^{\frac{-rn}{m}}$ is indeed a threshold function for the property B.

Corollary 1 If n is a fixed positive integer, then a threshold function for the property that ${\omega }_r$ has an n-element independent set is $q^{-r}$ .

Corollary 2 If m is a fixed positive integer exceeding two, then a threshold

function for the property that ${\omega }_r$ has an m-element circuit is $q^{\frac{-r\left(m-1\right)}{m}}$ .

Corollary 3 If n is a fixed positive integer, then a threshold function for the property that ${\omega }_r$ contains a submatroid isomorphic to $PG\left(n-1,q\right)$ is

$q^{\frac{-rn\left(q-1\right)}{q^n-1}}$ .

To show that the preceding three results are valid, we are required to check that the appropriate submatroids are k-balanced. For example, in Corollary 1, the n-element independent set must be k-balanced; this is the free matroid $U_{n,n}$ . Corollary 2 requires one to verify that an m-element circuit is k-balanced; this is precisely the uniform matroid $U_{m-1,m}$ , while in Corollary 3, the projective geometry $PG\left(n-1,q\right)$ needs to be k-balanced. For a more thorough discussion of this material, the reader is referred to Proposition 2 and Theorem 5 in [2] .

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	White, N. (1986) Theory of Matroids. Cambridge University Press, New York.
[2]	Oxley, J. (1992) Matroid Theory. Oxford University Press, Oxford.
[3]	Al-Hawary, T. (2007) A New Class of Matroids. African Diaspora of Math, 4, 87-91.
[4]	Al-Hawary, T. (2017) Certain Classes of Fuzzy Graphs. European Journal of Pure and Applied Mathematics, 10, 552-560.
[5]	Al-Hawary, T. (2002) Characterizations of Certain Matroids via Flats. Journal of Automata, Languages and Combinatorics, 7, 295-301.
[6]	Al-Hawary, T. (2001) Characterization of Matroids via OFR-Sets. Turkish Journal of Math, 24, 1-11.
[7]	Al-Hawary, T. and McNulty, J. (2001) Closure Matroids. Congressuss Nemerantuem, 148, 93-95.
[8]	Al-Hawary, T. (2004) Closure Matroid Properties. Mu’tah Lil-Buhuth Wad-Dirasat, 19, 35-43.
[9]	Al-Hawary, T. (2003) Feeble-Matroids. Italian Journal of Pure and Applied Mathematics, 14, 87-94.
[10]	Al-Hawary, T. (2000) On Balanced Graphs and Balanced Matroids. Mathematical Sciences Research Hot-Line, 4, 35-45.
[11]	Al-Hawary, T. (2001) On k-Balanced Matroids. Mu’tah Lil-Buhuth wad-dirasat-Natural and Applied Sciences Series, 16, 15-22.
[12]	Al-Hawary, T. and Horani, B. (2016) On Product Fuzzy Graphs. Annals of Fuzzy Mathematics and Informatics, 12, 279-294.
[13]	Al-Hawary, T. and Horani, B. (2017) On Product Intuitionistic Fuzzy Graphs. Italian Journal of Pure and Applied Mathematics.
[14]	Narayanan, H. and Vartak, M. (1981) On Molecular and Atomic Matroids. In: Combinatorics and Graph Theory, Vol. 885, Springer, New York, 358-364.
[15]	White, N. (1992) Matroid Applications. Cambridge University Press, Cambridge.
[16]	Erdös, P. and Rényi, A. (1961) On the Strength of Connectedness of a Random Graph. Acta Mathematica Academiae Scientiarum Hungaricae, 12, 261-267. https://doi.org/10.1007/BF02066689
[17]	Kelly, D. and Oxley, J. (1981) Threshold Functions for Some Properties of Random Subsets of Projective Spaces. The Quarterly Journal of Mathematics, 32, 463-469.