Matrices and graphs theory and applications to economics

Tiziana Calamoneri and Rossella Petreschi Silvana Stefani and Anna Torriero Luigi Guido Ceccarossi A Comparison of Algorithms for Computing the Eigenvalues and the Eigenvectors of Sym

Matrices and Graphs

Theory and Applications

to Economics

Matrices and Graphs

Theory and Applications

Dipartimento di Matematica "Guido Castelnuovo"

Universita di Roma "La Sapienza", Italy

World Scientific Publishing Co Pte Ltd

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MATRICES AND GRAPHS

Theory and Applications to Economics

Copyright © 1996 by World Scientific Publishing Co Pte Ltd

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In the meantime the editors organized another conference on «Matrices and Graphs: Theory and Economic Applications», held as the previous in Brescia during June 1995, partly with different invited lecturers The confer-ence was a success again and therefore the first editor applied to the Italian National Research Council and got a second contribution, that came only re-cently

While the lecturers of the first conference, who were not at the second one, were a bit upset, having submitted their paper without seeing any proceedings published at that time, the lecturers common to the first and the second con-ference suggested to join the contributions and publish a unique book for both conferences This is what we did

During all these years, both authors were very busy lecturing, researching, publishing, raising more funds to make their research possible Most papers arrived late and were carefully read by the editors, then the search of suitable referees was not easy, so that the reviewing process took also a while, some papers being sent back to the authors for corrections and then submitted again

to referees A complete re-editing was necessary in order to get the uniform editor's style , well, these are the reasons of such a delay, but eventually here

we are

The book reflects our scientific research background: for academic and entific reasons both of us were drawn to different research subjects, both shift-ing from pure to applied mathematics and statistics, with particular attention

sci-to data analysis in many different fields the first edisci-tor, and sci-to operational research and mathematical finance the second So, in each of the steps of this long way, we collected a bit of knowledge

The fact that in most of investigations we dealt with matrices and graphs suggested us to investigate in how many different situations they may be used

This was the reason that led to the conferences; as a result, this book looks like a patchwork, as it is composed of different aspects: we submit it to the readers, hoping that it will be appreciated, as we did

In fact, the numerous contributions come from pure and applied ematics, operations research, statistics, econometrics Roughly speaking, we can divide the contributions by areas: Graphs and Matrices, from theoretical results to numerical analysis, Graphs and Econometrics, Graphs and Theoret- ical and Applied Statistics

of a sign stable matrix, based on some properties of the eigenvectors ated to a sign semi-stable matrix Szolt Thza in Lower Bounds for a Class

(0,1) matrices It is interesting to note that the problem can be formulated

in terms of a semicomplete digraph D, if one wants to determine the smallest sum of the number of vertices in complete bipartite digraphs, whose union is the digraph D itself Tiziana Calamoneri and Rossella Petreschi's Cubic

graphs of degree three, and at most cubic graphs, i.e graphs with maximum degree three and show a few applications in probability, military problems, and financial networks Silvana Stefani and Anna Torriero in Spectral Proper-

of graphs through the spectral structure of the associated matrices and on the other how to get information on the spectral structure of a matrix through associated graphs New results are obtained towards the characterization of real spectrum matrices, based on the properties of the associated digraphs Guido Ceccarossi in Irreducible Matrices and Primitivity Index obtains a new upper bound for the primitivity index of a matrix through graph theory and extends this concept to the class of periodic matrices Sergio Camiz and Yanda Thlli in Computing Eigenvalues and Eigenvectors of a Symmetric Ma-

method for computing eigenvalues and eigenvectors of a symmetric matrix, to more classical procedures Divide et Impera is used to integrate those proce-dures based on similarity transformations at the step in which the eigensystem

of a tridiagonal matrix has to be computed

Among contributions on Graphs and Econometrics we find Sergio Camiz paper I/O Analysis: Old and New Analysis Techniques In this paper, Camiz compares various techniques used in I/O analysis to reveal the complex struc-

ture of linkages among economic sectors: triangularization, linkages son, exploratory correspondence analysis, etc Graph analysis, with such con-cepts as centrality, connectivity, vulnerability, turns out to be a useful tool for identifying the main economic flows, since it is able to reveal the most impor-tant information contained in the I/O table Manfred GilU in Graphs and

a system of equations and with necessary and sufficient conditions for this lution to hold He shows how through a graph theoretic approach the problem can be efficiently investigated, in particular when the Jacobian matrix is large and sparse, a typical case of most econometric models Manfred Gilli and Giorgio Pauletto in Qualitative Sensitivity Analysis in Multiequation Models

s0-perform a sensitivity analysis of a given model when a linear approximation is used, the sign is given and there are restrictions on the parameters They show that a qualitative approach, based on graph theory, can be fruitful and lead

to conclusions which are more general than the quantitative ones, as they are not limited to a neighborhood of the particular simulation path used Mario FaUva in Hadamard Matrix Product, Graph and System Theories: Motivations

struc-ture can be handled by using Hadamard product algebra, together with graph theory and system theoretical arguments As a result, efficient mathematical tools are developed, to reveal the causal and interdependent mechanisms asso-ciated with large econometric models At last, International Comparisons and

application of graph theory to the analysis of the European Union countries based on prices, quantities and volumes Graph theory turns out to be a most powerful tool to show which nations are more similar

Gio-vanna lona Lasinio and Paola Vicard in Graphical Gaussian Models and

mer-its of regression and graphical modelling approach are described and compared both form the theoretical point of view and with application to real data Francesco Lagona in Linear Structural Dependence of Degree One among

a linear structural dependence among data, thus deriving a particular vian Gaussian field Bellacicco and Tulli in Cluster identification in a signed

and graphs, by including clustering into the wide class of a graph tion in terms of cuts and insertion of arcs to obtain a given topology

transforma-After this review, it should be clear how important is the role of matrices and graphs and their mutual relations, in theoretical and applied disciplines

We hope that this book will give a contribution to this understanding

We thank all the authors for their patience in revising their work A special thanks goes to Anna Torriero and Guido Ceccarossi for their constant help, but especially we would like to thank Yanda Tulli, who did the complete editing trying to (and succeeding in) making order among the many versions

of the papers we got during the revision process Last, but not least, thanks

to Mrs Chionh of World Scientific Publishers in Singapore, whom we do not know personally, but whose efficieny we had the opportunity to know through E-mail

October, 1996

Sergio Camiz and Silvana Stefani

The manuscripts by Sergio Camiz, Guido Ceccarossi, Manfred Gilli, vanna lona Lasinio and Paola Vicard, Francesco Lagona, and Bianca Maria Zavanella, referring to the first Conference, have been received at the end of

Gio-1993 The manuscripts of Antonio Bellacicco and Yanda Tulli, Tiziana moneri and Rossella Petreschi, Sergio Camiz and Yanda Tulli, Mario Faliva, Manfred Gilli and Giorgio Pauletto, John Maybee, Silvana Stefani and Anna Torriero, and Szolt Tuza, referring to the second Conference, were received at the end of 1995

Cala-This work was granted by the contributions from Consiglio Nazionale delle Ricerche n A.I 94.00967 (Silvana Stefani) and Consiglio Nazionale delle Ricerche n A.I 96.00685 (Sergio Camiz)

Sergio Camiz is professor of Mathematics at the Faculty of Architecture of Rome University ~La Sapienza~ In the past, he was professor of Mathemat-ics at Universities of Calabria, Sassari, and Molise, of Statistics at Benevento Faculty of Economics of Salerno University, and of Computer Science at the American University of Rome He spent periods as visiting professor at the Universities of Budapest (Hungary), Western Ontario (Canada), Lille (France) and at Tampere Peace Research Institute of Tampere University (Finland), contributed to short courses on numerical ecology in the Universities of Rome, Rosario (Argentina), and Leon (Spain), held conferences on data analysis ap-plications at various italian universities, as well as the Universities of New Mexico (Las Cruces), Brussels (Belgium), Turku and Tampere (Finland), and

at IADIZA in Mendoza (Argentina), contributed with communications to

vari-ous academic congresses in Italy, Europe, and America After a long activity in the frame of computational statistics and data analysis for numerical ecology, and in programming numerical computations in econometrics and in applied mechanics, his present research topics concern the analysis, development, and use of numerical mathematical methods for data analysis and applications in different frames, such as economical geography, archaeology, sociology, and political sciences He was co-editor of two books, one concerning the analysis

of urban supplies and the other on pollution problems, and author of several papers published on scientific journals

Silvana Stefani is a Full Professor of Mathematics for Economics at the University of Brescia She got her Laurea in Operations Research at the Uni-versity of Milano She has been visiting scholar in various Universities in War-saw (Poland), Philadelphia (USA), Jerusalem (Israel), Rotterdam (the Nether-lands), New York and Chicago (USA) She has been Head of the Department of Quantitative Methods, University of Brescia, from November 1990 to October

1994 and is currently Coordinator of the Ph.D Programme ~Mathematics for the Analysis of Financial Markets~ She was co-editor of two books, one concerning the analysis of urban supplies and the other on mathematical meth-ods for economics and finance, and author of numerous articles, published in international Journals, in Operations Research, applied Mathematics, Mathe-matical Finance

AUTHORS ADDRESSES

Universita di Teramo

Via Cruccioli, 125, 64100 Teramo, Italia

Universita "La Sapienza"di Roma

Via Salaria, 113, 00198 Rama, Italia

E-mail: Calamo@dsi.uniroma1.it

Universita "La Sapienza"di Roma

P.le A Moro, 2, 00185 Roma, Italia

Universita Cattolica di Milano

Via Necchi 9, 20100 Milano, Italia

Universita "La Sapienza"di Roma

P.le A Moro, 2, 00185 Rama, Italia

E-mail Iona@pow2.sta.uniroma.it

Appli-cate

Universita "La Sapienza"di Roma

P.le A Moro, 2, 00185 Rama, Italia

John Maybee University of Colorado

265 Hopi PI Boulder, Co 80303 USA

Universita "La Sapienza di Roma "

via Salaria, 113, 00198 Roma, Italia

U niversita Cattolica di Milano

Via Necchi 9, 20100 Milano, Italia

E-mail Torriero@aixmiced.mi.unicatt.it

Universita di Brescia

Contrada S Chiara 48/b, 25122 Brescia, Italia

Hungarian Academy of Sciences

1111 Budapest, Kende u 13-17, Hungary

e-mail tuza@lutra.sztaki.hu

Universita "La Sapienza"di Roma

P.le A Moro, 2, 00185 Roma, Italia

Universita Statale di Milano

Via Visconti di Modrone, 20100 Milano, Italia

E-mail Zavanell@imiucca.unimi.it

Tiziana Calamoneri and Rossella Petreschi

Silvana Stefani and Anna Torriero

Luigi Guido Ceccarossi

A Comparison of Algorithms for Computing the Eigenvalues

and the Eigenvectors of Symmetrical Matrices 72 Sergio Camiz and Yanda Tulli

I/O Analysis: Old and New Analysis Techniques 92 Sergio Camiz

Manfred Gilli

Qualitative Sensitivity Analysis in Multiequation Models 137 Manfred Gilli and Giorgio Pauletto

Hadamard Matrix Product, Graph and System Theories: Motivations

Mario Faliva

International Comparisons and Construction of Optimal Graphs 176 Bianca Maria Zavanella

Graphical Gaussian Models and Regression

Giovanna Jona Lasinio and Paola Vicard

Linear Structural Dependence of Degree One among Data: A

1 Introduction

We deal with n x n real matrices Such a Matrix A is called semistable (stable)

if every A in the spectrum of A, u(A) lies in the closed (open) left-half of the complex plane The real matrix sgn(A) = [sgn aij] is called the sign pattern of

we let Q(A).be the set of all matrices having the same sign pattern as A We also will write A in the form A = Ad + A where Ad = diag[an, a22, ,ann]

and A= A-Ad

Let u be a complex vector u = (Ul' U2, , un) We say that U is

Ad, then U is q-orthogonal to Bd for every matrix BE Q(A)

Then A is called combinatorially symmetric and we may associate with A the

i # j and aij # o The graph G(A) is a tree if it is connected and acyclic We

also use, for any matrix, the directed graph D(A) defined in the usual way The real matrix A is called sign semi-stable (sign-stable) if every matrix

in Q(A) is semi-stable (stable) We will deal only with the case where A is irreducible in order to keep the arguments simple (Gantmacher, 1964) All of our results can be readily extended to the reducible case

We will prove the following results about sign semi-stable matrices Theorem 1 The following are equivalent statements:

2 Matrix A satisfies

(ii) aijaji ::; 0 for all i and j, and

(iii) every product of the form ai(1)i(2)ai(2)i(3) ai(k)i(l) = 0 for k 2:

{l,2, ,n}

0, i = 1,2, , n such that the matrix DAD- 1 = Ad + S where S is skew

Theorem 2 The following are equivalent statements about a sign semi-stable

matrix:

I' The matrix A has A = 0 as an eigenvalue

Theorem 3 The following are equivalent statements about a sign semi-stable matrix:

I" The matrix A does not have a purely imaginary eigenvalue

2" No matrix in Q(A) has a nonzero purely imaginary eigenvalue

Ad

The equivalence of conditions (1) and (2) of Theorem 1 is a well known result due to Maybee, Quirk, and Ruppert (see Jefferies et al, 1977 for one proof of this result) All the known proofs of this equivalence make use of one

of the classical stability theorems By proving that (1) ::::} (2) ::::} (3) ::::} (1) we can avoid the use of any stability theorem, a fact of some independent interest

A consequence of Theorem 1 is that the family of sign semi-stable matrices can be identified with the family of matrices of the form A = Ad + S, where

S is skew-symmetric and A satisfies (i),(ii), and (iii) This fact is used in an essential way to prove Theorem 3

Our proofs of Theorems 2 and 3 lead directly to simple algorithms for testing a given matrix satisfying the conclusions (i),(ii), and (iii) to determine whether or not it is sign-stable

Finally, given Theorems 1,2, and 3 we can state the following sign stability result

Theorem 4 The real matrix A is sign stable if and only if the following four conditions are satisfied

(i) ajj ::; 0 for all j;

(ii) aijaji ::; 0 for all i and j;

(iii) every product of the form ai(1)i(2)' ai(2)i(3) ai(k)i(l) = 0 for k 2: 3

where {i(l), i(2), ,i(k)} is a set of distinct integers in N = {I, 2, ,n}

(iv) the matrix A does not have an eigenvector q-orthogonal to Ad

Suppose first that the matrix A is sign semi-stable The fact that (i),(ii), (iii), are then true follows by a familiar continuity argument which we omit Hence (1) ::::} (2) Given that (2) is true and A is irreducible it follows that,

if aij :f 0 then aji :f 0 also For suppose aij :f 0 and aji = O Since there

is a path from j to i in A, (iii) is violated Thus A is combinatorially metric But then G(A), the graph of A exists, is connected and has no cy-cles Hence G(A) is a tree Then by a theorem of Parter and Youngs (1962),

sym-there exists a positive diagonal matrix D such that DAD- 1 = Ad + S where

Ad = diag[all' a22, , ann], S = [Sij], with Sii = 0 for i = 1,2, , n, and

Sij = -Sji for all i :f j Thus (2) ::::} (3) Now set A = DAD-l and

sup-pose Au = AU Taking scalar products on the right and left with U yields

u·Au = U· Adu+u· Su = U· AU = ~lul2 and Au·u = Adu·u+Su·u = Alul 2

We have U· AdU = Adu· u and U· Su = -Su· u Hence 2Adu· u = (A + ~)luI2

eigen-an eigenvector of A belonging to an eigenvalue on the imaginary axis, then we

must have Ui = 0 if i E Io by (1), i.e u is q-orthogonal to Ad

Note also that it follows from the proof of Theorem 1 that, if aii = 0, i =

1 n, A is skew-symmetric and all the eigenvalues of A are purely imaginary,

hence A is not sign stable If aii < 0, i = 1 n, then every nonzero vector

U satisfies Re() ) < 0 so A is sign stable Hence the interesting case for sign stability is 1 :5 1101 < n, which we assume to hold henceforth

3 The proof of Theorem 2

Let A be a sign semi-stable matrix By conditions (i) and (ii) every term in the expansion of det A has the same sign Therefore if det A = 0, it must

be combinatorially equal to zero and hence every matrix in Q(A) also has determinant equal to zero It follows that (1') implies (2') Our task is to discover when there exists a non-zero vector U such that Au = O Now U must vanish on the (nonempty) set 10 so we partition the components of a candidate vector U initially into the sets Z(Io), N(Io) where Z(Io) = {i liE Io}, Ui = 0

if i E Z(Io), and Ui =J 0 if i E N(Io) Now given a set Ip ;2 10 and a partition

of the components of U such that Ui = 0 if i E Z(Ip) and Ui =J 0 if i E N(Ip)

We look at the equations

Hence we must place k E I p+1 ' We do this for each such occurrence Thus

IpH ;2 Ip and Z(Ip) ~ Z(Ip+l) , N(Ip) ;2 N(IpH)' If the system (2) contains

no equation having only a single non-zero term, then IpH = Ip and Z(IpH) = Z(Ip) , N(Ip+l) = N(Ip) We will examine this case below Suppose that

Ip+l = N Then Z(IpH) = Nand u=O, i.e no matrix in Q(A) has zero as

an eigenvalue It remains to consider the case where we have some Ip = IpH

with IIpl < n so Ui = 0 for i E Z(Ip) and Ui =J 0 for i E N(Ip) Clearly every equation in system (2) at this point contains either no non-zero terms

or at least two non-zero terms We have N(Ip) ~ 2 and the induced graph

(N(Ip)) is a forest We claim that this forest consists of isolated single trees, i.e

S(N(Ip)) = O For suppose (N(Ip)) has a nontrivial tree To This tree has a vertex of degree one and there would then exist an equation in the subsystem

S(N(Ip))u = 0 having exactly one nonzero term, a contradiction Next let

IN(Ip) I = q and suppose there exists r rows in the subsystem L SijUi =

uEN{Ip)

0, i E Z(Ip), having two or more non-zero entries We have r ~ 1 so the set

of such rows is nonempty Let this set be Zo(Ip) and consider the submatrix

S(N(Ip))UZo(Ip) The graph of this submatrix is a forest on the q+r vertices

If I Zo (I p) I ~ q then the numbers of edges in this forest is at least 2r ~ r + q, a

contradiction Similarly, there cannot be two directed paths from vertex k to

vertex l in the directed graph D(So) where So is the matrix of the subsystem (2) for i E Zo(Ip) It follows that the subsystem Sou = 0 uniquely determines one or more eigenvectors U belonging to A = o Hence each matrix in Q(A)

has at least one eigenvector U belonging to A = 0 and vanishing upon the set

Z(Ip) ~ 10 Thus Theorem 2 is proved

4 The proof of Theorem 3

We look for an eigenvector U such that U is q-orthogonal to Ad and Au = iJL,

for some JL -=I o As in the proof of Theorem 2, we partition a candidate vector

u initially into the sets Z(Io), N(Io) with Ui = 0 if i E Z(Io) and Ui -=I ° if

i E N(Io) Now given Ip ;2 10 and a partition of U into Z(Ip) and N(Ip) we look at the equations

If any equation in the subsystem (3) contains exactly one nonzero term,

we have SjkUk = 0 and, as in the proof of Theorem 2, we adjoin each such k to

Ip Similarly, if any sum on the left hand side of an equation in the subsystem (4) contains no nonzero term, we have iJLuj = 0 and this contradiction compels

us to add the index j to Ip Doing this for every such occurrence produces the new set IpH ~ Ip and thereby the new partition, Z(IpH), N(IpH) If the subsystem (3) contains no equation having a single term and the subsystem (4) contains no empty sums, then Ip+l = Ip We examine this case below Suppose that Ip+l = N Then Z(IpH ) = Nand U = O Since every matrix

in Q(A) has the same zero-nonzero pattern, it follows that no matrix in Q(A)

has a purely imaginary nonzero eigenvalue

It remains to consider the case where we have some Ip = IpH with IIpl < n,

so Ui = 0 for i E Z(Ip) and Ui -=I 0 for i E N(Ip) At this point every equation

in the subsystem (3) has either no nonzero tenns or at least two nonzero terms Also the induced subgraph (N(Ip)) is a forest and contains no nontrivial trees, since every sum in the subsystem (4) contains at least one term This forest must contain at least two trees, because if it were a single tree the subsystem (3) would have an equation containing exactly one nonzero term Moreover, if

a tree in the forest is adjacent to a vertex j E Z (I p) there must also be another tree in the forest adjacent to vertex j for the same reason We must therefore

have IN(Ip)1 2:: 4 Let jo be the index of a row in subsystem (3) containing

q 2:: 2 nonzero terms Then vertex jo in G(A) must be adjacent to q distinct trees in the forest (N(Ip))

Now choose a pair of trees in (N(Ip)) adjacent to vertex jo Set Uk = 0

re-spectively Then the submatrices S(TI ) and S(T2) are disjoint nonzero skew

symmetric submatrices of S Hence they have nonzero purely imaginary values iJ.LI and iJ.L2 Let VI and V2 be nonzero vectors satisfying S(TI)VI =

eigen-iJ.LI VI, S(T2)V2 = iJ.L2v2 Then any vectors aVI and f3v2 also satisfy these

equa-tions where a and f3 are nonzero constants If J.LI = J.L2 then we choose a and f3 to satisfy

where kl is a vertex in one tree and k2 is a vertex in the other Thus aVI and

J.LI #- J.L2 then choose ao such that aoJ.L2 = J.L2 and modify S by multiplying

an eigenvalue

This proves that some matrix in Q(A) has an eigenvector q-orthogonal

Parter S and J W T Youngs, 1962 «The symmetrization of matrices

by diagonal matrices~ J Math Anal Appl., 4: p 102-110

LOWER BOUNDS FOR A CLASS OF DEPTH-TWO

R x 0 the set of the IRI 101 entries lying in the intersection of those rows and columns We prove that if Rl, , Rl and 01, ,Ol are l sets of rows and l sets

In this note we investigate an extremal problem on a class of m by m 0-1

matrices, motivated by switching theory The particular case in question can

be formulated in several equivalent ways, as follows

• Suppose that the square matrix M = (aij) E {o,l}mxm has zero onal, and at least one of aij and aji is 1 for all i f j, 1 :::; i, j :::; m

diag-Minimize the total number of rows and columns in a collection of 1-cells

(submatrices with no 0 entry) such that each aij = 1 occurs in at least one of those I-cells

• Let B = (X U Y, E) be a bipartite graph with vertex classes X =

{Xl, , xm} and Y = {Yl , Ym}, and edge set E, such that XiYi ~ E

for aliI:::; i :::; m, and at least one of XiYj and XjYi belongs to E for all

if j, 1 :::; i,j :::; m Find the smallest total number IV(Bl)I+·· .+IV(Be)1

of vertices in a collection of complete bipartite subgraphs Bi C B, Bi =

El U···uEe =E

• Given a semi-complete directed graph D = (V, E) on m vertices, without

loops and parallel edges (i.e., each pair x, Y E V is adjacent either by just one oriented edge, or by precisely two oppositely oriented edges

Xy, YX E E), determine the smallest sum of the numbers of vertices in

complete bipartite digraphs Di C D (with all edges oriented in the same direction between the two vertex classes in each D i ) whose union is D

• Suppose that a circuit has to be designed with inputs Xl, ,Xm and

outputs YI, ,Ym, where a set of conditions Cij prescribes whether there

exists a directed path of length 2 from Xi to Yj (written as Cij = 1;

otherwise we put Cij = 0) Assuming Cii = 0 for alII::; i ::; m, and

Cij = 1 or Cji = 1 (or both) for all i '" j, 1 ::; i,j ::; m, minimize the

number of links (adjacencies) in such a circuit

The equivalence of the matrix problem and the two types of graph oretical formulations is established by the corresponding adjacency matrices:

the-In the bipartite case we define aij := 1 if and only if Xi is adjacent to Yj; or, conversely, we join Xi to Yj if and only if aij = 1 For digraphs, the entry

aij = 1 of the matrix corresponds to the edge oriented from vertex i to vertex

j

To see that the switching circuits also give an equivalent formulation,

no-tice first that each link involved in a path of length 2 verifying Cij = 1 for some

pair i,j either starts from an input node or ends in an output node Now, each

internal node Zk of a length-2 path connects a set Xk of inputs with a set Yk

of outputs, and the number of links incident to Zk is IXkl + IYkl Therefore,

Xk x Yk must be a I-cell in the 0-1 matrix (Cij) Conversely, each I-cell R xC

of r rows and c columns in a 0-1 matrix M can be represented by an internal node Z connected to r input nodes and c output nodes in the circuit to be constructed

Notation

We denote R xC := {aij I ri E R, Cj E C}, where R ~ {r}, ,rm } is a set of rows and C ~ {CI,"" em} is a set of columns (We may also view the 0-1 matrices as subsets of {rl,' , r m} X {c}, , em}.) The shorthand

l

U (Rk x Ck) = M means that the entry aij has value 1 in M if and only if

k=l

ri E Rk and Cj E Ck for some k, 1 ::; k ::; i (and aij = 0 otherwise) The

complexity, IT(M), of M is defined as

where the value of i is unrestricted

Obviously, the definition of u(M) can be extended for arbitrary (not

nec-essarily square) 0-1 matrices, but in this paper we do not consider the more general case; i.e., M E {O, 1 }mxm will be assumed throughout

is the case, for example, in the following two particular sequences, as proved

by Tarjan (1975)

Theorem 1 If m = 2 n and M = (aij) is the upper triangle matrix (aij = 1

Theorem 2 If m = (L n /2 J)' where Lx J is the lower integer part of x, i e the

Our main goal is to show that the lower bound of m log2 m in Theorem 1 is valid for a much larger class of (m x m) matrices Namely, we will prove the following result:

Theorem 3 If an (m xm) matrix M = (aij) (aij E {O, I}) has zero diagonal

Theorems 1 and 3 are best possible in general, as it is discussed in the ing section On the other hand, we are going to observe that the complexity

conclud-a(M) of a typical member of the class of matrices involved in Theorem 3 is much larger than O(mlogm) To formulate this assertion more precisely, de-note

Mm := {M = (aij) E {O, 1}7nX7n I aii = 0, aij + aji > 0 for j =f i} ,

M~ := {M = (aij) E {O, l}7nX7n I aii = 0, aij + aji = 1 for j =f i} Theorem 4 There is a constant c > 0 such that

3 The general lower bound

The subject of this section is to prove Theorem 3, i.e., that a(M) 2: m log2 m holds for all matrices M = (aij) of order m with zero diagonal, containing at

least one nonzero entry in each {aij, aji}, i =f j

Suppose that an optimal collection of all-l submatrices Rk x Ck C M

(i) Ai n Bi = 0 for alII:::; i :::; m, and

(ii) Ai n B j =f 0 or Aj n Bi =f 0 for all i =f j, 1:::; i,j :::; m

Let us recall now the following inequality from Tuza (1987)

Lemma 5 Suppose that {(Ai, Bi ) 11 ~ i ~ m} is a collection of pairs of finite

where the choice for Xk is done independently of those for all the other elements

of X For i = 1,2, , m, denote by Ei the event

The condition (i) implies that the events Ei are nonempty More explicitly, by the random choice of Y, we have

for aliI:::; i ~ m Moreover, the simultaneous occurrence of two events E i, Ej

would imply

and hence

Ai U Aj ~ X \ (Bi U Bj)

would follow This possibility is excluded by the condition (ii), however, fore

there-Prob{E + + Prob{E 1,

completing the proof of (1)

By what has been said above, the conditions of Lemma 5 hold for the sets

A, B i Consequently, putting p = 1/2 we obtain

4 The bound for almost all matrices

(4)

In this section we prove Theorem 4 The argument will be quite similar for

Mm and M~, therefore we can handle these two classes together

Instead of counting, we are going to select a matrix M at random from the corresponding class, and show that

lim Prob (a(M) < cm2/logm) = 0, m-+oo

provided that the value of C is chosen appropriately The probabilistic model for Mm is

Prob(aij = 1/\ aji = 0) = 1/3,

Prob(aij = 0/\ aji = 1) = 1/3,

Prob(aij = 1/\ aji = 1) = 1/3,

for each pair i,j (1 ~ i < j ~ m) independently, while for M~ the sponding probabilities are

We claim that, with probability 1-0(1) as m ~ 00, every I-cell RxC c M

satisfies

for some constant c' Indeed, denoting m' := c' log m, arbitrarily chosen m' rows and m' columns generate a I-cell with probability precisely

if they do not induce a diagonal element, because the presence of two dependent

entries {aij, aji} C R x C would also yield {au, ajj} C R xC Moreover, the probability to get a I-cell R x C is zero if a diagonal element is included On the other hand, the number of m' x m' submatrices is

therefore the probability that some of those submatrices contains no zero entry

Choosing c' := 2/ log (I/p), this probability will tend to zero, since m' -+ 00

as m -+ 00 Thus, all (m' x m') submatrices of M contain at least one zero entry, with probability 1 - 0(1)

Cij := IRkl + IGkl

For aij = 0, we simply define Cij := O Now we have

2 mm {Cij I aij = I}

m(m -1)

2c'logm

with probability 1 - 0(1) as m -+ 00 Thus, taking c = (2c')-I, the assertion follows

5 Concluding remarks and open problems

Finally, we discuss the tightness of the results proved above, and mention some related questions which remain open

5.1 Tightness of the lower bound m log2 m

Both Theorems 1 and 3 are tight, and in fact the upper triangle matrices of order m = 2 n involved in Theorem 1 are the simplest extremal examples for Theorem 3, too Let us denote them by Tn (where the order is 2n) Tarjan

(1975) proved the inequality

(6)

by an explicit construction A simple alternative way to prove (6) is to consider the following recursive procedure Clearly, for n = 1, {rl} x { C2} is the subma-trix required to decompose TI For n ~ 2, we can take RI := {r}, , r m /2}

and G I := {Cm/HI,"" Cm}, i.e., the I-cell generated by the first 2 n - 1 rows

and the last 2 n - 1 columns Then IRII + IGII = m = 2 n , and if we remove those 4n- 1 (nonzero) entries of RI x G I from Tn, the remaining nonzeros form

two triangle matrices isomorphic to Tn-I, which then can be decomposed separately, by induction

Consequently, denoting by s(n) the total number of rows and columns in the collection of I-cells obtained recursively, we conclude

s(n) = 2s(n - 1) + 2 n ,

from which s(n) = n 2 n follows

Many further examples can be given which also show the tightness of Theorem 3, but we do not have a characterization of those matrices

We should also note at this point that the structural description of the collections {(A, B i ) I 1 :::; i :::; m} of set-pairs attaining equality in Lemma 5

is another interesting open problem for further research

5.2 Other types of well-structured matrices

It would be worth investigating in a greater detail what kinds of structural properties of a matrix imply small or large complexity The results above illustrate how some conditions imposed on the pairs of entries of M can restrict the range of a(M) As regards relationships between pairs of rows or columns, the class of Hadamard matrices is one of the interesting examples to consider

A lower bound is given in Tarjan (1975), but as far as we know, the exact value

of a(M) has not yet been determined for those matrices

It is quite natural to ask how a(M) changes if the probability p occurring in the proof of Theorem 4 takes different values More precisely, suppose that

M = (aij) is an (m x m) matrix with zero diagonal, and let each entry aij

(i =I- j) be equal to 0 or 1 at random, independently of the other entries (or possibly depending just on the value of aji), by the rule

m -+ 00, provided that p(m) ::; 1 - {j (for an arbitrarily fixed {j > 0, and for all sufficiently large m ~ mo)

On the other hand, according to Theorem 2, for p = 1 we have

a(M) = (1+o(1))mlog2m

Hence, writing p in the form p( m) = 1 - q( m), the speed of the convergence of

q( m) to zero determines the expected asymptotic value,

a(m) = a(m,p) := E(a(M))

of the complexity of M The current methods are not strong enough to describe the exact relationship between q(m) and a(m), and we do not even know how quickly q(m) must approach 0 to ensure a(M) = O(mlogm)

Note that a(m,p) is small also in the case where p(m) itself tends to zero

at a sufficiently large speed From this point of view it would be interesting

to see which pairs of small and large probabilities (tending to 0 and to 1, respectively) yield the same asymptotics for the expected value of a(M)

5.4 Circuits of depth 2

The problem for 0-1 matrices is equivalent to the depth-two circuit problem

only if paths of length precisely 2 have to connect the prescribed input/output

pairs On the other hand, allowing a link from the input directly to the output would mean that the corresponding weight IRkl + ICkl = 1 + 1 = 2 associated with a degree-2 internal node is reduced to Ij or, more generally, a star from an input node to a set of c output nodes (or from a set of r input nodes to an output node) has weight one smaller than that of the corresponding (r x 1) or (1 x c)

submatrix This change, however, does not influence the asymptotic behavior

of a(M), because there can be no more than O(m) such star configurations in any optimal decomposition (covering) of a matrix of order m into I-cells Acknowledgments

The research was supported in part by the OTKA Research Fund, grant

no 7558

References

Alspach, B., L.T OHmann and K.B Reid, 1975 «Mutually disjoint families of 0-1 sequences» Discrete Math., 12: p 205-209

Caro, Y and Z Tuza ,1991 «Hypergraph coverings and local colorings»

Katona, G.O.H and E Szemeredi , 1967 «On a problem of graph theory» Studia Sci Math Hungar., 2: p 23-28

Perles, M.A , 1984 «At most 2 d +1 neighborly simplices in Ed » Annals

Tuza, Z , 1994 «:Applications of the set-pair method in extremal graph theory~ In P Frankl et al (eds.), Extremal Problems for Finite

hyper-Sets, Bolyai Society Mathematical Studies Vol 3, Janos Bolyai Math

Soc., Budapest, p 479-514

Tuza, Z , 1996 «:Applications of the set-pair method in extremal problems,

II.~ In D Miklos et al (eds.), Combinatorics, Paul Erdos is Eighty, Bolyai Society Mathematical Studies Vol 2, Janos Bolyai Math Soc.,

Budapest, p: 459-490

CUBIC GRAPHS AS MODEL OF REAL SYSTEMS

T CALAMONERl, R PETRESCHI

Dipartimento di Scienze dell'InJormazione Universitd di Roma "La Sapienza"

In this paper we deal with cubic graphs, i.e regular graphs of degree 3, and with

at most cubic graphs, i.e graphs with maximum degree 3 We recall two basic transfurmation techniques that are used to generate these graphs starting from a smaller graph, either cubic or general Moreover we show some applications To complete this brief survey we present the state of the art of a specific problem on these graphs: their orthogonal drawing

1 Introduction

Any system consisting of discrete states or sites and connections between, can

be modelled by a graph This fact justifies that they are a natural model for many problems arising from different fields

For instance, the psychologist Lewin proposed (Lewin, 1936) that the life

space of a person can be modelled by a planar graph, in which the faces

rep-resent the different environments

In probability, a Markov chain is a graph in which events are vertices and

a positive probability of direct succession of two events is an edge connecting

the corresponding vertices (Hoel et al., 1972)

Military problems like mining operations or destruction of targets may be

led to the maximum weight closure problem (Ahuja et al., 1993)

Different processes such as manufacturing, currency exchanges, translation

of human resources into job requirements find as natural models networks, i.e directed weighted graphs (Evans and Minieka, 1992)

This interpretation is also applied to financial networks, in which nodes represent various equities such as stock, current deposits, certificates of deposit and so on, and arcs represent various investment alternatives that convert one type of equity into another

The search of the solution of problems in so different fields justifies the existence of many types of graphs and many basic notions that capture as-pects of the structure of graphs Moreover, many applications require efficient algorithms that operate above all on graphs

In this paper we deal with cubic graphs, i.e regular graphs of degree 3, and with at most cubic graphs, i.e graphs with maximum degree 3 We recall two basic transformation techniques that are used to generate these graphs starting from a smaller graph, either cubic or general Moreover we show some

applications To complete this brief survey we present the state of the art of a specific problem on these graphs: their orthogonal drawing

In all this paper we use the standard graph theoretical terminology of Hartsfield and Ringel (1994)

2 Cubic graphs

A graph G is said regular of degree k or k-regular, if every vertex of G has

degree equal to k A graph is called cubic if it is regular of degree 3 When the

degree of the vertices is less than or equal to 3, we have a more general class:

at most cubic graphs (see Figure 1)

Figure 1: An at most cubic graph and a cubic graph

It is to notice that to restrict the problem to cubic graphs makes sometimes the solution easier to be found than in the case in which the problem holds for

at most cubic graphs On the other hand there are problems for which this restriction does not help

For example, let us consider the chromatic index problem (CIP) and the minimum maximal matching problem (MMMP)

CIP: "Given a graph G = (V, E) and an integer K, can E be partitioned into disjoint sets E 1 , • ,Ek with k ::; K such that for 1 ::; i ::; k, no two edges

in E share a common endpoint in G 7"

MMMP: "Given a graph G = (V, E) and an integer K, to decide if there

is a subset E' of E with IE'I ::; K such that E' is a maximal matching of G."

The first problem is open for at most cubic graphs (Garey and Jonhson, 1979) while it is polynomially solvable for cubic graphs (Johnson, 1981) The second problem is proved to be NP-complete by a transformation from vertex cover for cubic graphs and it remains NP-complete for at most cubic planar graphs and for at most cubic bipartite graphs (Garey and Jonhson, 1979) The orthogonal drawing that we present in the last section is an example

in which the more general problem is related to at most cubic graphs On the contrary, the regularity of the degree is fundamental when a graph is the model of an interconnection network, as we show in subsection 3.1

The first time that cubic graphs appeared in the literature was in an formal manner in Tait (1878) and in a more formal way in Petersen (1891) dealing with factorizations of graphs and related colourings

in-Many specific theoretical results on cubic graphs are known, but they require a background that is not possible to give here For a deep insight on this topics, see Ore (1967), Hartsfield and Ringel (1994) and Greenlaw and Petreschi (1996)

In a cubic graph the number n of vertices is always even and the number

of edges is 3n/2 If the cubic graph is plane, the number of faces is 2 + n/2

In the following we just recall two basic transformation techniques that are used to generate cubic (or at most cubic) graphs starting from a smaller graph, either cubic or general

The following construction method is due to Johnson (1963) and is based

on the concept of H-expansion We call H-graph the graph with 6 vertices and

5 edges shown in Figure 2

Figure 2: H-graph

Let G = (V,E) be a cubic graph on n vertices and let el = (V2,V4) and

The H-expansion of G, with respect to el and e2 is obtained by eliminating

el and e2 and adding two vertices VI and V6 with edges:

or by (V6, Vt}(V6, V2) (V6, V3)( VI, V4) (VI, V5)

Theorem 1 (Expansion theorem)

a connected cubic graph on n vertices

In Figure 3 the 8-vertex cubic graphs derived from a 6-vertex one is shown, when edges (V2,V3) and (V4,V5) are removed

Figure 3: A 6-vertices cubic graph H-expanded into two 8-vertices ones

The second method we consider allows to transform general graphs into cubic ones and it is given in Ore (1967)

Let G = (V, E) be a graph on n vertices and let ni be the number of vertices of G with degree i

The cubic-trasformation of G is obtained by enclosing each vertex V having degree not less than 4 by a circle, small enough not to intersect neither any other circle nor any edge-crossing Each intersection of the circle with an edge emanating from V will become a dummy vertex in G (see Figure 4)

Theorem 2 (Cubic Transformation):

n2 + n3 + 4n4 + 5n5 + + (n - l)nn-I vertices

Figure 4: Scheme of transformation by general to cubic graphs

This transformation is sometimes useful to solve problems on general graphs by utilizing properties on cubic graphs An example is the case of the four colour problem: "every map (or equivalently planar graph) is 4-colourable."It can be proved that this conjecture is true for any map if it

is true for any planar cubic graph (Ore, 1967: 117-118) If this cubic map is 4-colourable, the coloring of the original graph G is obtained simply by the contraction of all the circles introduced in the cubic transformation

In the next section, we will present some problems that have as ral model a cubic or at most cubic graph and that utilize the just presented transformation techniques

We will present only three different applications that cover different fields and seem particularly significant

is connected, but the number of connections out from the same processor is limited by physical characteristics

There are many useful examples of graphs which are used for tion networks with limited degree like the tree connection and the d-cube con-nection In particular, we present the Cube-Connected-Cycles network (GCC)

communica-introduced in Preparata and Vuillemin (1981) This network is modelled by

a cubic graph derived from a hypercube whose all nodes have degree d To obtain the eee model, the cubic transformation is applied to the hypercube,

as shown in Figure 5 when d = 3

Figure 5: A CCC interconnetion network of dimension 3

The cubicity of the eee allows to present it as a feasible substitute for other networks both for its efficiency and for its more compact and regular VLSI layout

3.2 Interaction of particles

Let PI and P2 be two sub-atomic particles having two trajectories from Xl to

X2 and from YI to Y2, respectively Let X and Y be the points of the trajectories

in which the particles interact because of magnetic attraction or repulsion The graph obtained by connecting Xl and X2 to X, YI and Y2 to Y, and X to Y is an H-graph (Figure 6)

The H-expansion techniques allow to show (Bjorken and Drell, 1994) that the interaction between different sub-atomic particles can be modelled by a general cubic graph

3.3 The mosaic problem

Mosaic problem is related to biology, chemistry and graphics in general It

consists in covering the plane with copies of the same shaped polygon The only regular polygons that can be used in a mosaic covering of the plane are

Figure 6: Interaction between two particles

hexagons, squares and triangles (Ore, 1967) It is easy to see that hexagons induce a graph that is cubic except along the border of the external face (Figure

7)

We want to conclude this paper presenting the state of art of a lar problem related to cubic graphs In view of the fact that the graphical representation of a graph is not unique, (see e.g Figure 8) and that a "good"-drawing may be either a starting or arriving point of different problems, the next section will survey orthogonal grid drawing of at most cubic graphs

particu-4 Efficient drawing of at most cubic graphs

An orthogonal drawing of a graph G = (V, E) consists in a graphical

represen-tation of G on a grid such that:

- vertices are represented by points and they lie on the crosses of the grid;

- edges are represented by a sequence of alternatingly horizontal and vertical segments running along the lines of the grid

Notice that the definition of orthogonal drawing limits G to be a graph of maximum degree 4

A point of the grid where the drawing of an edge changes its direction (from horizontal to vertical or vice versa), is called a bend of this edge

We call such a drawing an embedding if no edges have intersections different