FINDING A BASE FOR A VECTOR SPACE OF

POLYNOMIALS bY

LUISA INIARIA

MCOSIA MC ALLISTER 104 Comenius Hall

MORAVIAN COLLEGE

1200 Main Str.

BETHLEHEM

PA 18018

USA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

te1.6 10-86 1- 1405 fax6io-a6i-39ao-

email; mcallist2moravian. edu ....................................... ABSTRACT.

In [Z], and in other presentations this author

proposed the definition offuzzy graph as a

pair of vector spaces. n e underlying reason

for the departure flom its usual definition as depicting afizzy relation, is not only the

elegance ofthe new definition but also

because of the practicality of having the tools

of linear algebra available should we be able

to find a basis for each vector spaces.

fuzziness is portrayed in the form of numbers

then to find a base, the work in [l, 41 is

helpfil. What, ifwe wish to use functions

such as polynomials?

It is possible to show that under suitable

conditions, a polynomial can serve as a@zy

number[61.

In (31 and [ I ] we find also help to solve the

problem.. This work illustrates how the task

can be done.

Why is this research important? Its importance

relies on the fact that vector spaces are

spanned by their bases.If these bases are

known, then any vertex not Qnctioning

properly,or any link that is defectvive or

inactive can be made fully functional by

expressing them as a linear combinatiom of

the elements of its basis.

INTRODUCTION. As expected, before

strong results are achieved, there is some

theory to be presented thus some previous

results have to be at least mentioned.

Since both the works in[7] and in[8] are based

on a technique seeking identification of

cliques,we shall begin by succintly reviewing

their definition.

DEFINITION. A set of vertices C in a graph is

called a clique if the subgraph generated by C

is complete( namely, every pair of vertices is

joined by an edge).

A clique which is not a subset of a larger one

is called maximal [2,page177].

The simplest method to find maximal

cliques is provided by the tree-search method

which is of high numerical complexity J91.

Both results in (71and in [8] have a consi

-derably smaller complexity, that is to say that

they differ in the order of the running time of

the algorithm

How do we decide which computational

method is best? Recall that there are two

major types. Either we are able to prove that

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we have : (a) polynomial, or (b) an

exponential estimate.

In the former case, a function of s, which is

called the problem size,namely a measure of

the amount of input, if the time t is a quadratic

function of s,then we say that the complexity is

can apply the algorithm to the support

set of the fuzzy edge set or to the

support set of the vertex set. In other

words, we must be able to argue that if e 1, e 2, ... , e n I

forms a basis for the support E of the 2 of order O(s ). fuzzy edge set E(f) then the

If the function is exponential it means that there exist constants c 1>0 and k

corresponding labelled edges form a

basis for the fuzzy edge set E(f). >0,c2 >O

and k 2>1 such that

An argument by contradiction will yield

the desired conclusion. c k l < f ( s ) < c 2 k 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . for all but a finite number of values of

s. [2,page 77 and ff..

The difference between methods is often

a personal preference, besides consi

-derations of lower computa-

tional complexity[91.

DEFINITION-- A function is said to be

of order g(s), or in symbols,

REFERENCES [ll-Anton H . Elementary Linear

AlPebra,Wiley, 1987.

[2l-Carre' B. Graphs and Netwo rks,

Clarendon Press, Oxford, 1979.

[3l--McAllister L.M.N. "Neural

Networks: A Simulation Technique

f(s) is O(g(s)), under Uncertainty"

if there exists a constant k such that Proc. of the 1992 NAF'IPS Conf,

Puerto Vallarta, Mexico, pp.555-563, f(s) 5 kgk)

for all but some finite,and possibly

empty, set of nonnegative values of s.

Finally, we consider a major question.

Is there a contradiction between the

definition of a fuzzy graph as a

labelled graph and the definition of it as a pair of vector spaces? The

question has already been considered in

[31 ,and it was given a negative answer.

What is the real advantage ? I t does

not matter whether the label are real

numbers or polynomial. If we wish to

use either171 or [SI, we can because we

1992; [41- Mc Allister L. M.N. "Fuzzy Intersection Graphs, " Int . Journal. of ComDuters and Mathematics with Applications. vol. 15, no.10, pp.871- 886,1988; [[5l--McAllister L.M.N."Graphs under Uncertainty :A Review and a Fuzzy Approach" IJnternational Journal of fuzzv Sets and Svstems. to appear ;

[GI-Stefanidis P.&Paplinski A.P.&

Gibbard M.J. Numerical Operations

with Polvnomial Matrices. Lecture

Notes in Control and Information

284

Sciences, Springer Verlag, (Thoma

M&M. &Wyner A, Eds.), no.171, 1992;

171- Taljan R.E.&Trojanoski A.E., "Finding a Maximum Independent Set I' SIAM J . Comput., vol.6,no.3, pp.537-

546, Sept. 1977;

[SI -Tsukiyama S&Ide M.&Shirakawa

I ."A New Algorithm for Generating

all the Maximal Independent Sets SIAM Journ.Compuf, 6, pp.505-

5,1977; [Q] - H.Wilf AlPorithms and CO mplexitv ,

Prentice Hall Inc., Englewood Cliffs,

N.J., 1986.

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