Existence and representation of natural arithmetic graphs

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

~01.24,No.6,pp.100-103,1984

0X1-5553/84 SlO.CQcO.00 ~1986 Pergamon Press Ltd.

EXISTENCE AND REPRESENTATIONOF NATURAL ARITHMETIC GRAPHS*

YU.G. GRIGOR'YAN

',"[II The existence of natural arithmetic graphs for any number of ribs C,,?] is proved. A lower bound is found for the number of possible representations of natural arithmetic graphs. The representation of simple networks, cycles, prisms etc. by natural arithmetic graphs is considered. The number-theoretic and geometric properties of arithmetic graphs are discussed in /l3/. In particular, according to /2/, all graphs can be divided into natural an unnatural arithmetic graphs. Questions of the representation of arithmetic graphs are linked with the following existence problem: it is required to show that, given a fixed number p of graph vertices, and given any number of ribs, p.Ospf natural arithmetic graphs is continuously filled by the number of ribs.

1. Basic concepts. with some modifications.

Let

For convenience, the basic concepts of /l-3/ will be repeated W(n,...,n,),M=(m,,...,m,) be two finite sets of positive integers.

G(.\:.N) is called an arithmetic graph rf G is a marked graph Definition 1. The pair with vertices from N and (n,. n,).?I,. n,s,V, is a rib when n,tn,=AI. The set M is called a generating ,,.+n,=m is the weight of the rib (n,,n,). The arithmetic graph is set, and the number denoted by C(.LI.V). Definition 2. The graph G is called natural arithmetic if it can be written Z....,p}lM), where Yc%R(s,4,...,2p-I). Consider the integer-valued function v(w), defined in the set 9x=(3.4,.,.,2/J-I] and having the form

where

[x] is the We know /2/ generated by the arithmetic graph

as

C(.V(l,

(1)

least integral part of I. that, for fixed p, given any m.=%Q, relation (2) gives the number of ribs the total number g of ribs of the natural element m,. Hence, if ill=(ml....,mr). is given by L q =

The domain

of variation

of the function

y

v(h)

v 1s obviously

2. The problem of the existence this problem, we first need two lemmas.

(3)

the integer-valued

interval

of natural arithmetic graphs.

I
To solve

,O and any fixed vI)of integer-valued Lemma 1. given any odd positive )-vim, / z-1iniz) =\ irri..l -=\‘i,.while m.,,m.,.m.,.m,.=W such that ~(nz, [p/21-1, there are four elements for ~,~=[p/z]there are three elements p, @+I). (p+2)=W such that y(p)=\(,J-ll -\tp-l)=(p/?]. of integer-v.t?wd interval I<\'< p>O and any fixed \'.I Lemma 2. Given any even positive \ lIJPlf I iw,,) 1, while [ii/Z]-1 , there are four elements m,:.m,:.m,,.rn,,~J such that a=-\i,ri, for ~,=[p/?] there is one element (~+i)cHl, where ~(p-1)=(p/2] Proof of Lemma 1. We first note that, by (2), given any m,-=ll, se,r !i 8, wtr nave v(m,i= \-(?(/)+I)-m,). Hence, to prove the first part of Lemma 1, it is sufficle::: 7~: ic:w chat T(lSl# l= m.,=m.,+i(m,, is odd) . V(m.,)=Vd for any w.,=zm.:
ii

nr,:=m,,+1.

such that v(m..)=V(m.:)=V,l. m,, is odd with 1-l. For this, we r:r:: Let us show that the lamma is true for ,=,,-I and obtain _______-___ *Zh.vychis1.Mat.mat.Fiz.,24,11,1751-1756,lgE; LOO

Ir,

where

:..>-:I slCes 31

: , -_

101

+2<,,,,.-‘)
,I‘,

iu‘eFind Y from (2) in the light of

nr.,+2=

m.,+2-1

v(nt.,r2)=

[

~

1

I

v(m.,+T?)= v(m.,+3)=

From

,nr.,‘2)

T

(6)

I.

(4), (5) and the fact that m..- 1

=

-+I

1

2

[

[F]

m,,is odd:

= vo+l,

7a)

= [ T] + izvo+j.

'7b)

which proves the first part of Lemma (7) we have v(n,.,+?)=v(m.,+?)=v,+i, The second part follows from a direct evaluation from (2) :

v(p)=

t

P-f yj-1

v(ptl)=

I

[-y-1 P

\‘(p+Z)=

1.

Lp-l 1

(8)

2

Slrce p 1s odd, we have [(P-1)/2l=[P/21,and hence we have from (8): ~(p,=v(p+l)-v(p+2)=[p/?] Using Lemmas 1 and 2, and (3), we write two Diophantine The proof of Lemma 2 is similar. eqmtlons respectively with p odd and even: :9a1

r‘+2f?+...+(~PI2l-l)f~~,?]--I+~P/2lflp,?~=q. f,,,?,E(@, 1.2.3); f,.fz....,f[p,2,-,~(o,1r2,3,4].

(9b)

f,+2f*+...+~[P/2]-1~f,P,?,-_(+[P/21f,P~?]=q. f,,f*r....f~P,2,--I~(0.1,2.3,1t), flp/?l= (0.1) Thi;lsthe problein of the existence of natural arithmetic abjlity of Diophantine equations (9) for every integer

graphs

reduces

to proving

the solv-

q=[@.C,‘].

Theorem. Numbers of type (9) with p odd and even respectively Integer-valued interval from 0 to C,z. The proof 1s by induction, intially for the case of odd p. Notice that, with odd SBO, we have C.z=C:_z+4[s/2]1. Ii is easily shown directly that the theorem holds for P=3. Assume then, by (9a), for any integer q=[O,Cf_,] we have

fll; the entire

that It

holds

close6

for

t,+2t,+...+([(r-2)/2l-i)t,,.~?,,l,_,+[(s-2)/2]t,,._l,,,,-q. I,.II.....ff(.-*),*I-,=P Since p LS odd, we can rewrite

1,2,3,41,

(10) PES-2;

(Ill

t,(._zr/?,=(O. I.3.R}

(11) as (12)

t,+2t,+...+([s/2]-2)f,“?~~?+([s/2]-I)L,,rl~_l=q. i[l,l,_,' (0.1.2.3) 1,.r,.....t~.,*~-2"(0,1.2.3.4). We shall prove the theorem

for

p=s.

i.e.,

show that numbers

of the type

t,+-2f,f...+((s/2l-l)t~.,?,-,+[s/2]t,.,~,=q. f,.1% ...>f[*/21-1=(0, 1,2,3,4). fill the closed integer-valued interval [O,C.*]. If hoids. If SbC?_?, then, by (101, we have

f,*,?]E(o.

1. 2. 3).

q=[O.C%_,1. by hypcthesls

q-(4[s/2]-I)=[O,C:--21. Hence,

by cl;),

(13), and in view of our assumption, t,+2t,+

. ..+([S/2]-2)f.,

the theorem (13)

we have

1,__~+([8/2]-l~I~.,?,_,=q-(11[8/?~--1~

(141

(0.1.2.3). ~I.~2r~..,~l‘/Z]--Z~~0,1. 2.3.4). f,.,?]_,= Ke rewrite We lntrodcce Then, using

(14) as f1+2t2+...+([s/21-2)f[,,*~-*+ ([s/2]~l)(t,,,~,'l)~3[P/~]=q.

(is)

I f,.,rl=3 'I.?--L=f,./?,-_(-I. f,'=f,.....f~./21--Z=f~~'21-2.

(16)

the notation (161,

we can rewrite 1,‘+2tz’+

(15) as +([s/21t2)f;,,2,_2 +([S/2]-l)f;.'Z,~,~f,:-~j[Si-?]= ',.

!lil

w,:.;ch ay (12) and 1161, It follows from (17) that f,'. fz',.... f~,,*]_-:.f;,~~]_, =((I ILL!‘! !) I,_ qF=(O, 1.23) proves the theorem for odd p. The proof is similar for even p. Nc:ice The solvability of Diophantine equations (9) follows obviously fr z the theorem. ti‘at expression (9a) with r,=4.fz=4...., fip.-:+-l=t. fIp :,=:1 is equal to C,'. Denote by o(q) the number of possible solutions of the Diophantine sqtiations for each flxed q. Then the number of solutions Slq)for all possible qs[(~_C;,'] ar.5 xl:,-p is qive;i, b; ~l:e theorem, by

102 2*p-s of all possible with g ribs. BY (181, S(q) 1s almost an order less than the number representations of the natural arithmetic graphs, obtained in accordance with definition and is a lower bound. 3. Representation of some classes of natural arithmetic is no general method of identifying and representing natural arithmetic interest ot obtain the arithmetic form of certain classes of graphs. Given the simple network with p vertices vIv?..."p-,"p. we shall prove: Proposition

1.

Every simple network

is a natural

arithmetic

2,

graphs. Since there graphs,

it is of great

(19,

graph which

can be written

as

c(\‘(l.2.....p)~M(p+l.p+3)), p even

(20aj

odd G(.\'(I.2,....p}~M(4.p+2,~+4)j. P

(20bj

The proof will be given constructively. be coded by the sequence

With p even,

(ij(P)(3j(p-2) (5)(P-4)

the sequence

of vertices

(19) can (20

(4)(p-1)(2)

and of alternating odd and even numbers; the number of odd numbers increases from 1 to p-1 of even numbers, decreases from p to 2. Then, by definition 2 of a natural arithmetic graph, the weights of the ribs of the graph are m,=p+l and m?=p+3. Since the vertices (21) are distinct and the total number of ribs is q=v(p+f)+v(p+3)=p-i, representation (20aj is a network. Case (20b), corresponding to odd p, is proved in a similar way. From Proposition 1 we have. Proposition

2.

Every simple cycle is a natural

arithmetic

graph,

which

can be written

as

(21aj

G(5(1,2....,p)~M(3,p+l,p+3)). p even G(S(~,~,...,PJIM(~,~P+Z,P+~)), P

(21b)

odd

The proof follows at once from the inclusion of element m=3 in the generating set of simple network (20); this ensures the appearance of rib (1, 2),theclosure of the network, and representation (22). It should be mentioned that the arithmetic representations obtained, of simple networks and cycles, are not unique even over the natural number set .~(!.2. .., and the question of all the possible natural representations P). remains open. p-2n vertices and Take the graphs G,.Gz (Fig.1) with 3rr-2 ribs, representing ribbon-shaped objects, developed respectively with an odd and an even number of square cells (without the broken arrows). Proposition

3.

Graphs

G,, Gz are natural

arithmetic

graphs

of the form G(N(1,Z....,2n)lM(Zn+l, 2n-1,2n+3)),

12;j

being graph G, ifn is even, and Ctif n is odd. The proof is constructive and is accompanied by direct coding of vertices of graphs G,,& by the broken diagonal arrows issuing from vertices 1 and 2n. Coding from vertex 1 is rea1izc.Z by a sequence of increasing odd numbers 1, 3 . . ..Zn-I Coding Fig.1 from vertex ?n is realized by the sequence of decrasing even 51,?!I-?...., numbers 2. Obviously, with n even, the number L'~I~I codes the vertex of the lower right corner, and 1 the vertex of the upper right corner of It is easily seen that, with this coding, all the vertical ribs 3f graph 6, and vice versa. will be generated by the element m,=2n+l, and the horizontal ribs, by the elements G,.G2 ml=",<-1,m3=2n+'l. This provides representation (23). By (3) the total number of ribs is y=~(?fr+l)+v(2r~-l)+v('Ln+~j=3n-2. Proposition 3 is thus proved. From it we obtain. Proposition

4.

Graphs

G,.(:; (Fig.1) are natural

arithmetic

graphs

G(.~(1.2.....2~1)~.ll(?n+l.2n-1.2n+3,3,4n-l)),

of the type

11;i CJ if R is even, and C. if n is odd. For, graphs C,,G, are obtained from G,.C? by the addition of ribs (1, 2) and (J,,_"t,-l~ ,,1.=3. "l,='tU--l. to the inclusion in their generating set of elements an3 which corresponds hence we have representation (24). With n even, CJ describes all the family of even n-faceti In particular, with n=4 we have the cube. The arithmetic representation of rhesc prisms. polyhedra is useful in connection with the important geometric interpretation of arithmetic graphs /3/. With n odd, C. describes a family of non-planar graphs, which are ina sense discrete analogues of the Mobius band. IIIparticular, C, with n=3 is the same as the familiar graph K1,,. It should be mentioned that, though the outer arithmetic representations of graphs (,.t, different are the same, the conditions imposed on the parameter R lead to qualitatively This suggests the existence of close number-theoretic cc:.::.. topological graph structures. tions between the arithmetic representations of a graph and its topological structurI being

REFERENCES 1. GRIGOR'YAN YU.G. and MANOYAN G.K., Topics in the arithmetic interpretation of undirected graphs, Kibernetika, No.3, 129-131, 1977. statistical properties of arithmetic graphs, 2. GRIGOR'YAN YU.G., Classificationand Kibernetika, No.6, 9-12, 1979. 3. GRIGOR'YAN YU.G., Geometry of arithmetic graphs, Kibernetika, No.4, l-4, 1982. Translated

U.S.S.R. Comput. Maths.Math.Phys.,Vo1.24,No.6,pp.103,1984 Printed in Great Britain

by D.E.H.

0041-5553/84 $lO.CO+O.OO 01986 Perqamon Press Ltd.

ON A PROBLEM OF NON-LINEAR PROGRAMMING FOR UNSMOOTH TARGET FUNCTIONS* * l

YU. V. ALEKSANDROV We ccnsider the problem of maximizing the maximum function ~(r)=mar(/‘(r)~l~(l.2. ...)v}). /,(J! are smocth functions, .r=!?cL‘,,. A similar problem is encountered e.g., when analysinq the guaranteed sensitivity of performance factors and the adequacy of systems of equations /I, L/ when the domain of deviations of the parameters and of the initial coordinate deviations is The feature of the present problem lies in determination at each iteration of the fixed. appropriate direction (in the non-linear programming sense), and in the rise in this direction not in general assuring convergence of the search algorithm. Two methods of successive approximations are proposed for finding the stationary points of any (not necessarily convex or concave) functions o(r) in a convex compact set Q; they are a development of the conditional gradient method /3/. Other ways of solving the problem are described in /l, 2/. to In the first method, at each point IA the lift is performed directly with respect several directions, found for the surfaces f.(z), I=R,(II). lying at a "depth" e from T(Q). including (I(.).The next approximation m(s), obtained as .I!,+,is chosen in the direction where a result cf the lift, is greatest. In the second method, at point I*.we choose among the f>(z). r~:lf,(iri. the surface ilh(z) for which the directional derivative 1s in a sense maximum i:! Q. In the resulting direction we perform a lift with respect to f,,(z) until at the end of the lift I((+)does not exceed $(~a). If this condition is not satisfied, e is split up and the procedure repeated, until, with decreasing E, we have for the first time. v(%+t)>viQ) The convergnece of these algorithms is proved. An important point is that, in both methods, the direction chosen for finding *I--I ma; not be appropriate; this I.S a feature of the methods. We consider the example when the search for max v(r) by the algorithms gives a result in circumstances when the ordinary conditional gradient method using motion along p(r) "jams." REFERENCES ALEKSANDROV YU.V., On guaranteed sensitivity of linear optimal systems, Izv. Akad. IJa-k SSSR, Tekhn. kibernetika, No.2, 168-113, 1975. ALEKSANDROV YU.V., Application of the method of Lyapunov functions to estimate the sensitivity of the vector performance factor of multiply connected systems, in: The direct method in stability theory and applications (Pryamoi metod v teorii ustoichivos:: i ego prilozheniya), Nauka, Novosibirsk, 1981. DEM'YANOV V.F. and RUBANOV A.M., Approximate methods of solving extremal problems (Priblizhennye metody resheniya ekstremal'nykh zadach), Izd-vo LGU, Leningrad, 1268. Translated

* * T!i‘?,cc-._zletetext is depcsited

at VINITI,

::0.1670-84, DEP.,

1’62

by D.E.B