Review 1. Personal Communication:

Mathematics Department Indiana University

April 19, 1986

Professors

M. N. S. Swamy and

K. Thulasiraman

c/o Prof. M. N. S. Swamy

Dean of

Engineering and Computer Science

Concordia University

Montreal, Canada

Gentlemen,

The recent profusion of books on Graph Theory has been such that I do not make a strong effort to look into each and every one of them when it appears. Most of them seem motivated by the thought "If there's any mathematics course that should appeal to the Computer Science students, Graph Theory is it; so there should be money in writing an 'Elementary Graph Theory' book. Since my book will be Elementary, I can write my book by the cut,- and-paste method out of the books I learned the subject from." I do not enjoy seeing new books of this kind coming onto the market.

It was therefore for me a very distinct pleasure when I ran across your book Graphs,Networks and Algorithms the other day, a book of fine scholarship, well-informed on recent developments in the subject whose exposition is of such good quality that it could also serve as a "beginners' text". It is also a surprise to see such a book written by a couple of engineers. (The usual failing of engineers as authors in mathematics is that they are unable to distinguish the mathematics from the physics--they think of an "electrical current distribution" as a laboratory concept and of Kirchhoff's Laws as "experimental facts " , blinded to the idea that a current can also be "defined " and Kirchhoff' s laws taken as "axioms"--a very necessary approach in order that one can see electrical network theory as mathematics which has applications going beyond electrical networks themselves!).

Let me congratulate you on a very fine job of scholarship and writing. I sincerely hope your book enjoys the popularity it deserves. I am learning something by reading it--for example, I have always seen the "Theorem of the Greedy Algorithm" mentioned in connection with the names of Rado and Edmonds, and had not realized that Gale and Welsh were also early discoverers of it.

I have a couple of (very minor) criticisms which touch on your treatment of my own work.

(1) Despite your obvious slant toward applications to electrical network theory, you have missed a great opportunity to point out the usefulness of some of the material in the context of "nonlinear networks". My paper "Solving Steady State Nonlinear Networks of 'Monotone' Elements" (I.R.E. Transactions on Circuit theory, Vol.CT-8, pp.99-104)is easily accessible to the general reader, and its material has been presented in standard works ( e .g., Berge and Ghouila-Houri, "Programming , Games and Transportation Networks" , Berge's "Graphs and Hypergraphs", Lawler's book which is well known to you, and Rockafellar's "Network Flows and Monotropic Optimization".) It seems a pity that you missed an opportunity to mention it.

You mention the book of Ford & Fulkerson as the "standard work" on a certain subject. It is by now quite obsolete; the above mentioned book of Rockafellar covers the subject far better (at least, from my point of view).

The "no-gain property of resistive networks", attributed by you to Wolaver 1970 (who apparently saw it in the context of linear networks) is in fact also valid for networks of nonlinear (monotone) resistances. For the case of a single current-source or voltage-source it appears in my 1960 paper "Monotone Networks" (Proc. Royal Society of London vol. A257, pp. 194-208), and the multi-source theorems are rather obvious to those who understand the method of proof given there (which is, to be sure, not as easy as Wolaver's method).

A sidelight on this subject which you may find of interest, although it is not directly relevant to your book, is that my work on nonlinear electrical networks of 1960-61 was the direct inspiration for the "Method of Monotone Operators", lately enormously popular in the area of Nonlinear Functional Analysis (with applications to nonlinear partial differential equations, nonlinear integral equations, etc.) For a reference, see: Some Topics in Nonlinear Functional Analysis, by Mohan C. Joshi and Ramendara K.Bose, Wiley/Halstead Press 1985, Chapter 3 ( particularly, p.40).

Please do not regard the above "criticisms" as serious ones. On the whole, I find your scholarship absolutely remarkable.

Very sincerely yours,

Sd.

(Prof.) George J. Minty

P.S. to Prof. Swamy: I have enclosed a copy of this 1etter for Prof. Thulasiraman. I think it will be more certain of reaching him if you would enclose it with your next correspondence with him.

Yet another postscript: I think the first publication pointing out the probable applicability of matroid theory to problems of electrical network synthesis was mine--see "On the Duality Principle of Electrical Network Theory" in P. Rosenstiehl (ed.), Theorie de Graphes (proceedings of a meeting held in Rome, Italy), Dunod (Paris) and Gordon & Breach (U.S.), 1967. I am happy to say that I was probably the instrument of introducing both Lawler and Weinberg to Matroid Theory. (In 1960-64, they were professors of Electrical Engineering at the University of Michigan, while I was in the Mathematics Department there.)